Like the sound of Seattle?

The boy on the corner of Pike Place and Pine Street knows his music history. Perhaps 13 years old – 14 at most – he cannot have been alive when the man printed on his chest died. Yet even on this dank Washington State morning – rain drifting in from the Pacific; tourists ambling down the slope, aiming for Pike Place Market – the face on the T-shirt still fits. Kohl-eyed, stubble-cheeked, wearing a quietly pained expression, it is an image that became ubiquitous – but whose fame grew and swelled here in Seattle.

Two decades ago, Kurt Cobain gazed out from magazine covers, TV screens, glossy posters and the stages of vast venues. And his arrival was swift. One minute the world knew nothing of Nirvana. The next they were everywhere, the planet's biggest band. Their coming, in 1991, altered the musical landscape and for three years, with Seattle as its epicentre, "grunge" ruled the airwaves – and never more so than in the shape of Nevermind, Nirvana's keynote album, released 20 years ago today.

Purists may point out that the band had already released their debut, 1989's Bleach – but Nevermind was the distillation of their art, angry yet accessible, loud yet loveable. And in its opening track "Smells Like Teen Spirit", it brandished what is still one of the most recognisable guitar riffs ever written.

The rest is so much newsprint: Cobain's 1992 marriage to fellow rock star Courtney Love – an alliance that pushed him from the music sections to the gossip pages; his struggle to cope with his new, vaunted profile; a discordant third Nirvana album (1993's In Utero); a spiral into heroin addiction; his violent suicide at just 27.

But what I want to discover when I land in an overcast Seattle is whether Cobain's story still echoes in the city that provided much of its setting. For Nirvana were a band as firmly pinned to America's most north-westerly metropolis as the Beatles to Liverpool, the Velvet Underground to New York or Elvis Presley to Memphis.

The parts of Seattle that Cobain inhabited in happier times are tricky to trace. The billboard above the Paramount Theater, where Nirvana played in the first flushes of Nevermind-mania in October 1991, is touting an upcoming Adele show. And its sister venue, the dilapidated Moore Theater, has seen better days – although that was already the case when Nirvana took to the stage of this 1907 relic in June 1989.

Elsewhere, The Crocodile – where Nirvana made a "surprise" appearance in October 1992 – sticks to the script, a small venue amid the noisy bars of the Belltown district. But The Vogue, a nearby club where the band tore through their first Seattle set in April 1988, has since become Vain, a beauty salon – though the long brick-walled interior that ebbs behind the wigs in the window betrays the building's past.

The heritage gap is now plugged by "Nirvana: Taking Punk To The Masses" – an exhibition at the Experience Music Project, one of the city's grandest cultural spaces. Put together with the assistance of Krist Novoselic, Nirvana's bass player and Cobain's childhood friend, this retrospective places the band within the context of the fertile music scene that emerged in Washington State in the late Eighties – while also displaying holy-grail totems from their brief lifespan.

"I literally went into Krist's attic and took down bins of stuff that he hadn't looked at in a decade," explains the curator, Jacob McMurray.

These include the bassist's personal set-list from the final Nirvana concert (in Munich on 1 March 1994), the acoustic bass he strummed at their celebrated MTV Unplugged in New York show in November 1993 – displayed alongside the Musrite Gospel guitar Cobain used to write much of Nevermind, and the band's first record contract, with the Sub Pop label.

But it is the more personal pieces that give the exhibition colour and texture: a thrift-store cardigan from Cobain's wardrobe, complete with holes in the sleeves where he worried them with his fingers; a photo of the band on the Dover-Calais ferry on their 1989 European tour, all windswept and bemused; a snap of a grinning Cobain recording demo tracks in 1986, an optimistic, hopeful 19.

Outside, the Experience Music Project tells a story of its own – shiny, rainbow-hued and visibly the work of architect Frank Gehry. Perched below the 605ft Space Needle, the city's 1962 vision of the future, it is an arty exclamation mark. And it is hardly alone.

For Seattle is a fascinating city, San Francisco's cool cousin, all implausibly steep streets and fractious climate – but blessed with a youthful vibe (it was only founded in 1852) where its Californian "neighbour" pines for the Sixties. It is also a curious hybrid, a hard-working port with a real cultural edge. And its two-tone appeal is immediately apparent – the sweat of its industrial zone, where the main Boeing plant churns out aircraft; the high-brow Seattle Art Museum, where water-colours sit next to striking contemporary pieces.

At times, these two facets merge: in the Olympic Sculpture Park, on the cusp of the Bay, where modern works (notably sculptor Alexander Calder's red metallic beast The Eagle) lie so close to the water that huge ships moor metres away as freight trains clank past; in Pike Place Market, where you can eat gourmet seafood (Sockeye salmon with mango salsa for $16.95/£11) at Athenian, or buy an art-print to go with your latte at coffee shop Local Color – but where fishmongers still eviscerate the daily catch on slick marble slabs.

Nirvana were part of this – the musical force that put Seattle on the map, and yet also symbolic of its character, forward-thinking and tireless of work ethic, always on tour.