Art rock darling St Vincent, aka Annie Clark, makes bold, complex records and has worked with the likes of Sufjan Stevens and David Byrne. But her songs have simple messages and, as the singer admits, she rarely suffers self-doubt

In the Manhattan restaurant, a beanie covering her hair, Annie Clark, 31, otherwise known by the grander sobriquet St Vincent, twists a thin black scarf this way and that, framing her face dramatically. It's all very "Greta Garbo, and Monroe, Dietrich and DiMaggio".

Clark's quiet-spokenness and low-key demeanour is jolting next to her music, which is bold, multi-instrumental and multi-layered: synths plink and plonk in dramatic ways, woozily distorted vocals jostle with heavy drum beats and jamming guitars. Her songs are gritty and rhapsodic: "I prefer your love to Jesus," she sings in the most moving song of her forthcoming, eponymous fourth album. She says it was for her mother, who has just recovered from serious illness. "I won't say what, but she's better now."

The New York Times, reviewing the tour for her third album, 2011's Strange Mercy, which peaked at No 19 in the US charts, noted Clark's "confounding" sound: "dry, compressed drums; neon synthesiser glare; aggressively processed guitars; calm, dispassionate singing". On stage, she throws herself about. It's not dancing, but she says, "violent and weird, a celebration and fight at the same time".

The intelligent and wry Clark, who will soon play concerts in London and Manchester, is the current darling of art rock, revered by fans for her consummate guitar-playing. Last year, she won the Smithsonian American ingenuity award for performing arts.

She named herself St Vincent after a line in a Nick Cave song that referred to the famous Manhattan hospital, St Vincent's, where Dylan Thomas died in 1953. (At one point in our conversation, she segues into a highly entertaining anecdote about how, a couple of years ago, terrified of what she assumed was a rat running from room to room in her apartment, she picked up a volume of Thomas's poetry and threw it at the creature, killing it on impact, "even though I don't like any cruelty being done to living things".)

Clark's fourth album goes right through the emotional wringer: one song, Psychopath, is about a date she once had with someone who is now a friend; other titles include Regret, Severed Crossed Fingers, Bring Me Your Loves and Every Tear Disappears. In person, though, she is less agonised than ethereal and sprite-like (last week, the online teen magazine Rookie posted a charming video of her showing off her mean soccer skills). "I have a sneaking suspicion that everybody has dark thoughts, but maybe doesn't say them," she says.

As she polishes off a plate of eggs and spinach, Clark, who lives in the East Village, says that she has just arrived home from Christmas with her family in Texas. "In New York, I have to repress the urge to call men 'sir' and women 'ma'am'." (She smiles, then shivers. "I don't want to be 'ma'am' to anyone.")

While Texas was a conservative place to grow up in, her family is not, "although being a freak in Texas is more difficult than being a freak in New York City where everyone is. I cultivated an inner world and a real obsession with music because when you're young that's your armour, how you form your identity."

Clark's mother is a social worker turned administrator for a non-profit organisation and her stepfather "works in the tax business". Her natural father lives in Tulsa, Oklahoma, and after her mother separated from him, she moved to Texas when Clark, the youngest of three girls, was seven. "My mother tried to insulate us as much as she could, but divorce is difficult for any child. One way or another you feel abandoned." Over time, her mother and father's new families have melded: she now has four brothers and four sisters. "My dad was a lapsed Catholic," she says drily. "He was more interested in Bertrand Russell than the pope."

At school, Clark admits people may have found her "a little odd", but that she was not "a figure of derision. I was always trying to get my sister to do fun, mischievous adventures, like to tip over the porta-potties (portable lavatories) at school or steal candy". She liked drawing and at five made her first guitar from cardboard and rubber bands. She made up songs and listened to classic rock and Led Zeppelin. Music "obsessed" her.

"When I was nine, Nirvana's Nevermind came out and that was a sea change across the world," Clark recalls, sipping her cafe au lait. "One day everyone was in polo shirts, the next in flannel. Suddenly my heroes were not C+C Music Factory, it was Kurt Cobain." She wonders, given how fragmented music distribution is now, and the proliferation of so much, so fast by the internet, whether something as culture-shifting as Nirvana could have the same profound and prolonged impact today.

Bands such as Sonic Youth and Pearl Jam became Clark's teenage favourites. "I remember my mum dropping me off at the CD store and me trying to judge by the guy behind the counter's words or look whether something was cool or not. 'Actually,' he'd say, 'you should check this out.'" Did her mother worry about her desire to become a musician? "No. It wasn't like I had a drug problem. They saw I was pretty obsessive. My mum's brother was a musician. They saw I had a proclivity for music, and wanted to show me the realities of life on the road, so they sent me on tour with my uncle."

Clark was 16 and her uncle was Tuck Andress (one half of the jazz duo Tuck and Patti), "a mythical figure" when she was young. On tour, her job was "to test the voltage of every outlet on stage so the gear didn't explode and that there were fresh flowers on stage and in the dressing rooms". She went to Japan and Europe. "I don't think I've ever worked so hard to this day."

Afterwards, Clark attended Berklee College of Music in Boston, Massachusetts, for three years before dropping out. "I wanted to move to New York and have fun. I wanted to cultivate my own instincts and write my own music, be my own front-person, rather than trying to be the best session guitarist in Los Angeles, which was what the school was trying to push me towards. They wanted everyone to be middle of the road, ready to play every potential style of music. I'm not anti-education, but my best education was listening to records and going out on the road."

But this bid for independence was derailed when she excitedly moved to New York aged 20, ran out of money in three months and moved back to Texas with her "tail between my legs". "Two or three weeks later", she joined the symphonic rock/pop band the Polyphonic Spree, playing guitar, and "I haven't had a day job since". In 2006, she joined Sufjan Stevens's band, before releasing her debut solo album, Marry Me, the following year.

Clark always wanted to be a solo artist. "My uncle told me when I was a teenager that if you want to be a person with confidence just pretend you are a person with confidence and eventually you will have confidence, so I did that. There's a fair amount of self-delusion involved. I never envisioned a Plan B. I'm almost immune to the idea of failure; it never occurred to me." She remains wary of working with others, though made an exception to collaborate last year with David Byrne on the much-praised album Love This Giant and subsequent tour.

With so many complex layers of instrumentation in her songs, is she a perfectionist? "There's a difference between perfection and being detail-oriented. Perfection is everything on time, in tune, not a hair out of place. That's vanilla. Detail-oriented is different: some songs are puzzles sonically. The producer and I slightly recontextualise sounds. We manipulate guitar sounds to sound like [other] instruments.

"I try to live at the intersection of accessible and lunatic. As far out as David Byrne and I go, we counterbalance that with a memorable melody or something for people to latch on to. I try not to put songs into ill-fitting clothes." And does her labyrinthine style owe anything to a love of classical music? "Stravinsky is my all-time favourite, his shit bangs," she assents heartily. "Like heavy metal and hip-hop, it grooves really hard."

She insists that however fraught her songs sound they are in fact "quite literal." Rattlesnake, on the new album, is about a walk she took by herself on a friend's ranch in west Texas. "It was a beautiful day and no one was around, so I thought, 'I'm going to take all my clothes off to fully experience this'. I was having this communion with nature when I saw this rattlesnake. I took off running and when I got home had a shot of tequila."

Another song, Huey Newton, named after the co-founder of the Black Panther party, relates what happened after she took a sedative to relieve jetlag while on tour in Helsinki. "If you take one and go to sleep, you sleep for 12 hours. If you take one and don't fall asleep, you're high. It's bananas. You're in that high state between sleepfulness and wakefulness. I had this hallucination that Huey Newton was in the room with me. We didn't talk about the Black Panther party. We just kind of communicated. We understood each other. I was as high as a kite."

She says she has been listening to Beyoncé's latest album. "What I love about it is that she's the biggest star in the pop world, where there is a lot of focus on youth, especially for the hot new female-whatever. And here is Beyoncé singing about basically loving her husband, having a baby and being a fabulous woman. She didn't come pouting with a lollipop and pigtails and be something she's not. I think that's very helpful for people to see."

Clark thinks "being a feminist by action speaks louder than arguing about semantics. Being a strong woman in the world is a feminist act. It's impossible to be a woman and not see misogyny, but I don't walk through life feeling like a victim. I've been very lucky. The strongest thing a woman can do is be successful, powerful and excel at whatever they choose to excel at."

While her songs are drenched in the trials of love and relationships, Clark is single. "I live an absolutely unconventional life where I travel 10 months a year, so it's not always the easiest to hold a relationship together. I didn't ever imagine my wedding day. Someday I'd like to have kids, but I don't have that burning desire to."

Abruptly, eggs and spinach polished off, she removes the beanie to reveal a surprising large, tangled mass of grey hair, totally different to her usual brunette style. She says was inspired by old footage of David Bowie "who had tried to dye his hair blond but it had gone orange and was awesome", and also by a blonde contestant on the last series of TV show The Bachelor, in which a group of women compete for a man's romantic attention. One of the women had one arm "and was understandably horrified to learn that as part of their date she had to jump off the side of the building". It's comforting to know that Clark, cool as she is, can be caught in the headlights of trashy reality TV just like the rest of us.

St Vincent is out on 24 February. St Vincent plays the O2 Shepherd's Bush Empire on 20 February and Manchester Cathedral, 21 February; ilovestvincent.com