It's 1979 and "an earthquake is erupting", but not in Jamaica's legendary Orange Street. The locus for this particular tectonic upheaval is north London, where Madness, a local sextet that is soon to become a septet with the addition of dancer/singer/trumpeter Cathal "Chas Smash" Smyth, have just cut their debut single at cubbyhole Pathway Studios for a mere £200. Showcasing their homegrown interpretation of ska, the upbeat Jamaican pop particularly loved by lead singer Graham "Suggs" McPherson, the band's first single is a joyous carnival of squawking sax, rollercoaster keys, lurching rhythm and a guitar hook that sounds like a car backfiring – all in tribute to ska pioneer Prince Buster, whose 1963 song Madness gave the band their name (and The Prince its B-side). Released on 2 Tone, the label run by Jerry Dammers of kindred spirits the Specials, The Prince was an instant hit, and soon found the Nutty Boys making the first of many appearances on Top of the Pops.

The band's debut album, One Step Beyond, followed in swift measure, cementing Madness’s ska credentials with its titular lead single – which opened with Smash’s legendary call to arms (and feet) on behalf of their "heavy, heavy monster sound” before taking another Prince Buster tune for a tumble on their bouncy castle. The LP’s other tracks reflected the more eclectic jumble of the group's inspired live act, from affectionate rock’n’roll pastiche and Egyptian exotica to eccentric vaudevillian storytelling and more thoughtful, sophisticated fare such as My Girl. Written and originally sung by keyboard player Mike Barson, My Girl was no retread of the old Motown classic, but a kitchen-sink tale of heartache and misunderstanding that played out like Robert Wyatt scoring The Likely Lads, with Suggs singing glumly of hour-long phone calls to his girlfriend that are packed with mardy silence, and his need to “stay in and watch TV on my own every now and then”. Affecting and honest, it was an early sign that there was more to Madness than wacky videos and hopscotch ska.

Their early following included a strong skinhead contingent – common among all the 2 Tone groups – that saw Madness facing unfounded accusations of affiliating with racists. It was exacerbated by naive comments Chas Smash made during an interview with NME, which he responded to on the pointed Don’t Quote Me on That. The second single off their second LP, Absolutely – and a distinct tonal shift from the naughty schoolboys romp of lead single Baggy Trousers – Embarrassment offered a subtle but powerful repudiation of racism and ignorance. Lee Thompson wove reactions from more bigoted members of his family to news that his sister was to have a mixed-race baby into a storming anthem of alienation, as the song’s protagonist is scolded for being “a disgrace to the human race”. Set to Barson’s downbeat honky-tonk piano and Thompson’s scowling sax, Embarrassment made genius pop of such weighty material.

The Nutty business was doing so well by 1981 that Madness relocated to Compass Point Studios in the Bahamas to record their third album, 7. But while you can take the boys out of London, the album’s closing track shows you can't take London out of the boys, and proved that home was still on their minds. A darkly comedic skank, A Day on the Town found Suggs taking unpaid bus rides to the capital for mischief and evocatively chronicling the delights of a summer’s afternoon in London, as dodgy touts corral “the tourists into their traps/ Taking their money, the shirts off their backs”. Invoking a little Ghost Town-style menace, Suggs’s yell of “riots in London!” signalled a turn into eerie, dubby waters, the song’s coda reprising the doomy vibe of 7’s masterly lead single, Grey Day.

Dave Robinson, of the band's label, Stiff Records, had a profound influence on his charges, directing many of their early iconic music videos and their witty, low-budget biopic, Take It or Leave It. Robinson badgered the reluctant band into committing this cover of the Labi Siffre ballad – a concert favourite – to vinyl. The delicate pizzicato strings, arranged by producers Clive Langer and Alan Winstanley, were a testament to Madness’ fast-growing sophistication, while Suggs’s deftly underplayed vocals lent the more flowery parts of the lyrics an earthy, moving honesty. Exactly a decade after the release of Siffre’s original, It Must Be Love reached No 4 in the UK charts (it also went top 40 in the US), to the relief of Robinson, who had promised to sign over ownership of Stiff Records to the group if the single wasn’t a hit. Months later, his insistence that another Madness demo, The Chemist Facade, needed a chorus before it could be released as a single led to the group’s first chart-topper, House of Fun.

In the liner notes for the 1986 singles compilation, Utter Madness, Chas Smash – now known as Carl Smyth – recalled that the roots for Our House lay in his idea that each member should write a song about their family backgrounds, “as all our families were so different. No one else bothered. I did.” Written in that familiar Nutty voice – a mixture of warm nostalgia, affectionate wit and subtle pathos – Our House was a homely portrait of Smyth’s youth, with father, mother, sisters and brothers all cramped in one chaotic abode, love and mischief abiding, and a bittersweet middle eight suggesting those familial ties might not always bind. It’s the kind of unpatronising portrait of working-class life that pop music doesn’t seem to deliver any more, though its sentiments – not to mention the swooning strings, Barson’s barrelhouse piano chords and Chris Foreman’s twangy guitar – gave Our House enough universal appeal to score the group their only top 10 hit in the US. It was later spun off into a stage musical.

This selection draws heavily from the band's faultless hits compilations, Complete Madness and Utter Madness. That is no slight on the group’s album tracks, but a sign that the Nutty Boys are still perhaps England’s greatest post-Beatles singles group. By their early-80s peak, their rate of production was enough to deliver regular non-album hits, including It Must Be Love, House of Fun, the infectiously ecstatic Wings of a Dove and this 1983 nugget. The Sun and the Rain was a tribute to precipitation, and what could be more English than that? Accompanied by an urgent, string-laden stomp, Suggs sang of “standing up in the falling down”, while Barson’s drolly hammered away with glee on a honky-tonk piano.

As the 80s wore on and the policies of the Thatcher government bit, the band found it hard to play the nutty clowns of yore and said they needed to write about what was happening to Britain. One song from this time, Victoria Gardens – which bleakly essayed the homeless community in London’s “cardboard city land”, taking explicit aim at Thatcher with the arch “she said it’s for the good of us all” – was planned as a single from 1984’s Keep Moving, but was nixed in favour of this more moving take on the subject. Suggs sang of a homeless man and woman who find love in the street. The song’s power comes from how the grim realities of the verse blossom into the chorus’ rich romance, like something out of Charlie Chaplin’s City Lights: “Walking round you sometimes/ Hear that sunshine/ Beating down in time with the rhythm of your shoes.” The accompanying video saw Suggs and his wife, singer Bette Bright, play the song’s protagonists, dancing with their carrier bags in front of Arlington House, a Camden Town hostel for homeless people.

Barson left the band following Keep Moving and moved to Amsterdam. Despite gems like the old-before-their-years melancholia of Yesterday’s Men, the cherishable Tears You Can’t Hide and a charming cover of Scritti Politti’s Sweetest Girl, the remaining members felt little affection for their 1985 Barsonless effort Mad Not Mad, while a subsequent album in 1988 as the Madness (now minus the rhythm section of Daniel Woodgate and Mark Bedford) did little to arrest the decline, and the group broke up. In 1992, however, they re-formed for the Madstock gigs in London's Finsbury Park – which, infamously, caused an actual earthquake in north London – and seven years later they returned to the studio, with Barson, for Wonderful. That rarest of things – a reunion album worthy of the legend – Wonderful maintained Madness’s proprietary blend of the lunatic and the earthbound, with lead single Lovestruck weaving addictive, affecting pop from alcoholism. But the highlight was when Suggs handed the microphone to Ian Dury, a huge influence on the band. Dury played a cockney hoodlum making his claim for the throne of the London underworld to the sound of a fairground ska lurch. It would be his final recording before he died in 2000.

Madness’ ninth album, The Liberty of Norton Folgate, was greeted by "best-of-their-career" accolades on its release in 2009. The praise wasn’t a respectful doffing of the cap to venerated national treasures. A loosely conceptual work immersed in the kind of vaudevillian London milieu in which Suggs had grown up, and rhapsodising the capital city, Norton Folgate was a lesson in how ambition need not suffocate a band’s innate playful tunefulness, and, what’s more, how a band so rooted in the joyful anarchy of youth could glide successfully into maturity without losing the very qualities that made them precious. Perhaps the peak of the remarkable second act of Madness’ career (though last year’s Oui Oui Si Si Ja Ja Da Da was also a keeper), Norton Folgate’s literate, lunatic sprawl was introduced by this 10-minute overture, a fine introduction to the larger body of work. If you ever loved a Madness tune (and shame on you if you haven't), you need to hear it.