In the last month, crunchy formative-years favorites Weezer have dropped a pair of new songs that represent an interesting duality for the band and its fanbase. Late October saw the release of “Thank God for Girls,” a manic theatrical number that packs just about every lyrical and melodic self-indulgence that frontman Rivers Cuomo has developed over the last 15 years into one chaotic over-production that needs a Genius annotation from the artist himself to approach comprehensibility. A week later, the band followed that with “Do You Wanna Get High?,” an impossibly old-school-sounding alt-rock chugger that’s as immediately satisfying — albeit in almost the exact same way — as many of the tracks on the band’s 1994 debut LP. In unveiling the songs back-to-back, the question to the band’s audience seemed implicit: Do you guys like who we are now, or would you really rather we just go back to doing the old stuff?

For many long-suffering Weezer fans, the answer will undoubtedly be the latter option. The conventional narrative with the band has always gone: The first two albums (1994’s self-titled “Blue Album” and ’96’s Pinkerton) are the unassailable classics — though it took the band’s five-year disappearance for Pinkerton to be recognized as such — while 2001’s Green Album (the band’s first without original bassist Matt Sharp) is also decent, if a little unremarkable coming off a half-decade’s rest. Then things drop off with the following year’s Maladroit, and never really recover — from then on, every album contains a handful of good songs at best, with the rest ranging from kinda lame to downright embarrassing, with the half-exception of last year’s more warmly received Everything Will Be Alright in the End. Cuomo never quite connected with his whole fanbase in the 21st century on the same level that he did in the ’90s, and if there’s a healthy chunk of devotees out there who hear “Get High” — likely the quartet’s most faithful early-days throwback yet — and ask “What the hell took you guys so long?,” it would hardly be surprising.

But there is a minority of fans out there — and obviously I’ll include myself in that tally — who’ve liked Weezer’s last 15 years just fine, and even prefer the fried-brain frenzy “Girls” to the traditional pleasures of “Get High.” Yes, the band swings and misses a good deal of the time, but the fact that they’re still taking such big rips at all is commendable, and their whiffs are more fascinating than 99 percent of other bands’ hits. Rivers is nothing if not his generation’s Brian Wilson, and while his own post-Pet Sounds period is as fraught with failures as Wilson’s late ’60 and ’70s were, it’s also as full of gems that confound and stun and and occasionally transcend in a way that’s utterly singular to their creator.

As embarrassing as some moments have been for Cuomo and Co., what would really have been embarrassing is if they spent the entire second act of their musical career trying to relive Act One — look no further than last year’s sour, pandering glory-days retread “Back to the Shack” for proof of that. Besides, if we were really worried about second-hand humiliation from Weezer, we never would have made it past “In the Garage” and “Across the Sea” in the first place. The cult of Weezer has never been about the band being in any way cool, it’s about them being awkward and corny and stupid in all the same ways that we were, about us looking to them in our lamest moments to hear them say “It’s OK to be terrible sometimes.”

And so: The 21 best Weezer Songs of the 21st Century. You might not know the words to all of them, and your high-school memory associations with many may be minimal at best, but they represent a body of work that’s kept Rivers Cuomo as one of this country’s most compelling songwriters long after “Buddy Holly” and “El Scorcho.” And that’s for all-time.

21. “The British Are Coming” (Everything With Be All Right in the End, 2014)

The most fun alt-rock historical retelling since Pavement’s “No More Kings,” sweeping and charming and totally anthemic. Kinda outta place smack in the middle of Everything Will Be Alright, but if it presaged an eventual full-length Revolutionary War-themed rock opera, we wouldn’t be mad — Cuomo’s gotta come with his own Most Lamentable Tragedy one of these days.

20. “Run Away” (Hurley, 2010)

The unfairly maligned Hurley album contained a handful of Rivers’ best songwriting collaborations, the most interesting of which may have been the Ryan Adams co-penned “Run Away.” The control-freak kindred spirits find common ground on one of the most chiming power-pop melodies to appear in a Weezer song, accompanied with some absolutely gorgeous backing harmonies and a sentiment that perfectly splits the difference between Adams’ romanticism and Cuomo’s insecurity: “I just want to run away from here with you.”

19. “Photograph” (The Green Album, 2001)

Perfectly representative of both the strengths and weaknesses the Green Album: Immaculately produced, preternaturally catchy, and unapologetically vacuous. Not the first or second-best song entitled “Photograph,” but certainly top five.

18. “Miss Sweeney” (The Red Album Deluxe Edition, 2008)

A spiritual sequel of sorts to ultra-endearing ’90s B-sides “Susanne” and “Jamie” — odes to the band’s A&R rep and lawyer, respectively — “Miss Sweeney” sees Cuomo clumsily macking on his assistant, nervously stuffing way too many syllables into his all-business verses about measurements and contracts before dropping the facade with the blue-skied chorus. The bridge is best of all, the frontman unleashing over power-ballad guitars, “Miss Sweeney, I got to admit the truth / I am totally head-over-heels in love with you!!” And yes, there is a real-life “Sweeney, baby”: longtime band assistant Sarah Kim, who co-wrote the tune.

17. “Turning Up the Radio” (Death to False Metal, 2010)

The 2010 odds-and-ends collection Death to False Metal was a better listen than it had any right to be, shuffling through likeable new wave pastiches (“Automatic”) and Strokes-y northern-soul swingers (“I’m a Robot”) and even climaxing with a surprisingly faithful (and successful!) cover of Toni Braxton’s “Unbreak My Heart.” But the set’s first track was likely best, an exhilarating power-of-song ode (“Let the music puh-layyyyyy!!“) that Cuomo wrote in collaboration with about a dozen of the band’s YouTube followers, the impressive coherence of which suggests that Rivers’ fans know him better than he knows himself.

16. “Heart Songs” (The Red Album, 2008)

Cuomo’s ultra-sentimental tribute to the ’70s and ’80s songs that form his musical core, as Weezer’s hits did for so many children of the ’90s. It’s undeniably mawkish, and at times factually inaccurate — Rivers supposedly wanted the song to stay memory-based and thus unreliable, fair enough — but when it gets past the Nevermind revelation of the bridge and into the “THESE ARE THE SONGS I KEEP SINGING!” burst of the final chorus, it’ll give the ol’ ticker a twinge, for sure.

15. “American Gigolo” (Maladroit, 2002)

The pounding drums and grinding metal guitars of the first track on Maladroit gave listeners the impression that the album was a much more dramatic left turn than it actually ended up being. Too bad: the hair-metal fascinations of the song’s verses colliding with its jangle-pop chorus provide a visceral kick that the largely formless rest of the album could’ve used a healthy amount more of.

14. “Do You Wanna Get High?” (Non-Album, 2015)

As much of a shame as it would be to see Weezer follow the rabbit hole further down into all-consuming mid-’90s nostalgia, it’s still pretty hard to argue with the fittingly intoxicating rush of Those Guitars combining with Those Dynamics and Those Wails. There’s a reason some fans have been waiting around 20 years to see if Weezer would finally deliver a song like this again, and the narcotic temptations of the lyrics give “High” a darkness and intrigue we never heard from Cuomo the first time around, thankfully keeping it from becoming cartoonish in its retro leanings.

13. “Hang On” (Hurley, 2010)

One of Rivers’ best classic pop songs, swiping strings from Jim Steinman, Wall of Sound production from Phil Spector, and a chorus hook from the Four Seasons for a full-bodied, three-chord crusher that absolutely should’ve been Hurley’s lead single. Michael Cera provides backing vocals and mandolin; not that it particularly matters to the song, but such cross-generational synergy between two greats of adolescent awkwardness should always be celebrated.

12. “Thank God for Girls” (Non-Album, 2015)

An absolute mess of hunger and horniness — cannoli, the Sears catalogue, and Adam’s caradamom-flavored rib all feature prominently — made sublime by a cathartic chorus and Cuomo’s peerless ability to articulate the feeling of your mind getting short-circuited by overpowering sexual attraction. In anyone else’s hands, it’d be totally disastrous; in Weezer’s hands, it’s still at least a little bit disastrous, but also funny as f**k and begrudgingly understandable.

11. “Troublemaker” (The Red Album, 2008)

The Red Album-opening mission statement of Rivers 2.0, concluding (somewhat literally) over thick RAWK riffing that the standards of songwriting no longer applied to him: “Who needs stupid books / They are for petty crooks / And I will learn from studying the lessons in my dreams.” Cuomo Unleashed is almost too much to handle at times, as he commands and then sings-along to a cheesy guitar lick, rhymes “beee-yooootch” with “kiiii-oddds,” and climaxes by counting down to the millennium by his lonesome because he’s “such a special guy.” But there’s something uniquely exciting — even a bit frightening — about seeing a songwriter making peace with his own mania, and offering his truest self up to the public in judgment. “There isn’t anybody else exactly quite like me,” claims Cuomo, and after “Troublemaker” the point was certainly beyond dispute.

10. “This Is Such a Pity” (Make Believe, 2005)

Weezer had so exhausted audiences with the first three singles off Make Believe — the fun-but-obnoxious anthem “Beverly Hills,” the utterly inexplicable “We Are All on Drugs,” and the tiring singalong “Perfect Situation” — that its fourth single barely made a blip. It’s a shame, as “Pity” was the best of the bunch, a driving, sonically streamlined new wave jam that served as a better Cars tribute than any of the band’s songs actually produced by Ric Ocasek. The lyric was one of Cuomo’s best, too, a simple (by his standards) lyric of two people in a relationship causing each other pain for no apparent reason that resulted in his most sadly relatable chorus since Pinkerton.

9. “Go Away” (Everything Will Be Alright in the End, 2014)

Rivers’ personality has come to so overwhelm the rest of Weezer in the 21st century that the band’s music has verged on being downright solipsistic at times. Good, then, to get another voice in there on occasion — as last year’s Alright did inspiredly with “Go Away, ” a duet with another self-obsessed alt-rock beach bum in Bethany Cosentino of Best Coast. Their voices sound spectacular together, and the two do a credible job of maintaining a “Sometimes Always”-type push-pull dynamic over sunsoaked guitars and Patrick Wilson’s wave-crashing drums. Nobody will ever undress Cuomo on record quite as succinctly as Cosentino does with her first two words here: “Stupid boy.”

8. “The Girl Got Hot” (Raditude, 2009)

On their most obviously Top 40-influenced album of the new millennium, Weezer got into the game of using the same sort of Gary Glitter schaffel swing that Katy Perry and Flo Rida rode to the top of the charts at the end of the ’00s. To the credit of them and co-writer/producer Butch Walker, it worked beautifully, as Cuomo’s enjoyably leering tale of a former junior-high classmate not being the Little Girl He Once Knew (“What used to mean a little now means a lot / Oh my goodness me, the girl got hot”) sounds absolutely massive over the “woah-oh“-assisted beat. Never released as a single, the song has gone on to a second life as a jock jam at certain NBA arenas, though fans probably don’t know the artist behind it any more than they could identify “Kernkraft 400″ by name.

7. “Don’t Let Go” (The Green Album, 2001)

The song that announced that the third Weezer album would absolutely not be like the first two, “Don’t Let Go” is as satisfyingly slick and flawlessly tuneful as any in the band’s catalog, but was missing that certain something — honesty, vulnerability, maybe just a bum note or two — that initially made them the cracked voice of a generation. Almost 15 years later, some fans still have trouble forgiving them, while others are too busy trying to get the “ooooh, woahhhh-woah” refrain out of their head to care one way or the other.

6. “Freak Me Out” (Make Believe, 2005)

One of the most coolly unnerving entries in the Weezer song catalog, its spooky guitar harmonics underlining the debilitating social anxiety of the chorus (“You really freak me out / I’m so afraid of you”) with suspect tenderness. And the song’s anticlimactically enigmatic bridge — “I’m going to try to improve my manners / Everyone, yes, everyone is my friend!” — as followed by the band’s best non-“Jonas” harmonica line, was one of the first signs that we were headed for much greater and more wondrous weirdness with Rivers than we ever could have imagined from the man behind “Hash Pipe” and “Dope Nose.” (It’s maybe about a spider, if that makes it more or less disconcerting.)

5. “Dreamin'” (The Red Album, 2008)

The centerpiece of the band’s best 21st-century album, a rollicking refutation of adulthood and responsibility that’s every bit as gleeful and disconcerting as such a sentiment should be coming from a 37-year-old with a wife and kid. Cuomo leans into the dark side of his carefree fantasies in the song’s percussion-less, childlike bridge, echoing temporary lead vocalist Brian Bell’s “I am running through the meadow” sighs before taking over to belt “And the angels in the heavens / They are wondering / WHY AM I SO GLAD?” It’s beautiful and disturbing in its Syd Barrett-ness, but the song resists easy interpretation by ending on the affirmative singalong “I don’t want to get with your program!” In the Weezerverse, such moments of paralytic immaturity are neither admirable or condemnable, merely inevitable.

4. “Smart Girls” (Hurley, 2010)

The best filtering of Rivers’ female-induced dementia into a proper pop tune, in this case a turbocharged romp with a sledgehammer chorus that makes the singer’s frayed croak of “Don’t you know you make me lose my mind?” the most rhetorical question in the band’s catalog. The song zips by like a bullet train, and with its grocery list of titular foxy intellectuals (“Deb-bie! Don-na! Ta-ti-a-na!“), it comes off as the adult-in-a-candy-store flipside to the groaningly anguished “Tired of Sex” from a decade-and-a-half earlier. That it shares a title with an even more bizarre solo concoction from Cuomo’s idol is almost too perfect.

3. “If You’re Wondering If I Want You To (I Want You To)” (Raditude, 2009)

Nostalgic, factually convoluted, hopelessly romantic but also strangely cynical (“There may come a day when we have nothing left to say”) and standoffish (“Make a move, ‘coz I ain’t got all night”), “Wondering” is just about everything you could want from a 21st-century Weezer love song. Oh, and supremely catchy too, of course, with its acoustic campfire stomp, drum-along percussion breaks, barbershop-quartet bridge, and absolutely seismic chorus. “The Slayer T-shirt fit the scene just right” is a line that could have appeared in a song by approximately zero other bands in music history, but it’s hard to imagine a better lead image here.

2. “Island in the Sun” (The Green Album, 2001)

A permanent vacation of a perfect pop song, as idyllic and quietly purgatorial as any Jimmy Buffet crowd-pleaser, but with lyrics that don’t suck. With its subdued backing vocals, unnaturally tranquil guitar hook, and despairingly content lyrics, the overwhelming feeling of “Island” is one of escape that’s impossible to find your way back from, yacht rock where the yacht gets lost at sea, never to return. As an increasingly distant Cuomo cries “We’ll never feel bad any more!” after the final chorus, the song reveals itself to be closer to a suicide pact than a romantic getaway, a “(Don’t Fear) The Reaper” for the surf-rock set. Rivers undoubtedly agrees with Lloyd Braun: Serenity now just means insanity later.

1. “The Greatest Man That Ever Lived (Variations on a Shaker Hymn)” (The Red Album, 2008)

Rivers Cuomo condensing his entire Smile period into one six-minute track, an 11-part symphony that encompasses rap, metal, punk, showtunes, classical, choral, and more, and yet somehow stands as the most coherent and definitive statement made by Weezer in their Scott Shriner-era incarnation. Here, Rivers extends his middle finger as far as it can go to anyone who thought the band’s first few ’00s LPs were lacking in ambition or personality, giving fans a borderline-lethal dose of uncut Cuomo. But jarring as the song’s disparate sections feel up against one another at first, after a couple of listens it comes off as no bumpier than “Bohemian Rhapsody,” as you give yourself over to its singular sense of musical narrative and outsized-but-impossibly-compressed ambition. Rivers’ message is clear throughout: I shouldn’t have to follow any of the rules that other songwriters of my age and stature are supposed to follow, because I can do things that none of them can do. Whether or not you agree with him, it’s the kind of on-record bravado and willingness to push into uncharted waters — while most of their peers have long settled into veteran complacency — that has continued to make the band worth getting into such arguments over. “No more words will critics have to speak”? Not in this lifetime, thank God.