The New Yorker critic Louis Menand, reflecting on TS Eliot's transition from radical modernist to arch-conservative, wrote in a review of the poet's letters: "He tried to shut the door on modern life. It was too late of course. He was the author of Prufrock and The Waste Land. He was already inside."

Eliot would not have loved pop music but pop music loves Eliot. Ninety years after the publication of The Waste Land, he remains the lodestar poet for ambitious songwriters. They rummage through his masterpiece's treasure chest of arresting phrases: the "violet hour" and "bodies naked on the low damp ground" quoted in the Sisters of Mercy's Floorshow, "April is the cruellest month" kicking off Hot Chip's Playboy or the "red sails" picked up by David Bowie on Lodger (Bowie told William Burroughs in 1974 that he'd "never read" Eliot but I suspect he got around to it).

Likewise 1915's The Love Song of J Alfred Prufrock. "Like a patient etherized on a table" is paraphrased by avowed Eliot fan Win Butler in Arcade Fire's We Used to Wait, "Do I dare disturb the universe?" became a song title for Chuck D, and "the Eternal Footman" crops up in Tori Amos's Pretty Good Year. "Alfred J Prufrock would be proud of me," declare Manic Street Preachers on My Guernica. And 1925's The Hollow Men lends its name to songs by Faust, Gravenhurst and Cocteau Twins. And on it goes: Genesis, Gentle Giant, King Crimson, Van Morrison, Rush, EMF, Crash Test Dummies, Okkervil River, the Clientele … "This music crept by me upon the waters."

But why Eliot, above all other poets? One simple reason is that he is widely taught in British and American schools and he impacts on the adolescent imagination with peculiar force. The Waste Land may be unfathomably complex but it is easy to love regardless of whether you understand it. The language is juicy and pungent, full of fire and rain, rivers and dust, birth and death – lots of death. I remember deriving a thrill of pleasurable dread from its sense of crisis and doom when I first read it as a teenager. Lines such as "I will show you fear in a handful of dust" or "This is the way the world ends/ Not with a bang but a whimper" (from The Hollow Men) would be at home on the back of a goth's leather jacket. Eliot offers a vivid grown-up take on a teenager's sense that all is not right with the world. At a difficult age you get the impression he's on to something terribly important, even if you're not sure what it is.

So the message resonates, in sometimes simplistic ways, but the medium also has much to teach songwriters. Just months after his death, on 4 January 1965, Bob Dylan's Desolation Row described "Ezra Pound and TS Eliot/ Fighting in the captain's tower." You can understand the appeal to a man attempting to blast open the language of rock'n'roll in a period of sociopolitical flux. Eliot told Virginia Woolf that Joyce's Ulysses, which he believed did in prose what The Waste Land did in verse, "destroyed the whole of the 19th century". They were twin responses to the shattered postwar world – vast collages of competing voices which declared that the old ways were dead and new language was needed. But that new language was built from the bones of the old: a dizzying mosaic of allusion, quotation, pastiche and impersonation, assembled from ingredients gathered everywhere from the ivory towers to the saloon bars, ancient Greece to modern London, and inviting endless interpretation. It attempted to encompass everything in a way that could mean anything, which is a decent description of Dylan's mission in Desolation Row. When Eliot, traumatised by the strain of composing The Waste Land, later dismissed it, he used a line you could imagine Dylan pitching to an earnest interviewer: "a piece of rhythmical grumbling."

The poem remains a catalyst for jolting songwriters out of their usual approaches because of the relentless ventriloquism referenced in Eliot's Dickens-quoting working title for The Waste Land, He Do the Police in Different Voices. The style enables you to open up a song's meaning by sliding between characters and perspectives: not a single broadcast but a radio impatiently flitting between stations. Neil Tennant applied the technique to 80s London on West End Girls with its "too many shadows, whispering voices" in an unreal city where "we've got no future, we've got no past".

Thom Yorke reached for it to evoke his own sense of dislocation and lurking horror on Paranoid Android's neurotic babble of unidentified voices ("Please would you stop the noise", "That's it sir you're leaving", "Off with his head") talking but not listening – "a heap of broken images", to quote The Waste Land.

Eliot's influence extends across the whole of PJ Harvey's Let England Shake, which pieces together voices and images from multiple decades and countries, and collapses all that history into a single ongoing commentary on war and nationalism. You often can't tell which lines Harvey wrote herself and which she took from existing sources; among other things The Waste Land's collage technique is a licence to borrow without shame. One blogger has pointed out the similarity between On Battleship Hill ("Jagged mountains jutting out/ Cracked like teeth in a rotting mouth") and The Waste Land ("Dead mountain mouth of carious teeth that cannot spit"). The critic Donald Childs believes Eliot was actually referring to Gallipoli (the home of Battleship Hill), where his close friend Jean Verdenal had been killed in action. During the campaign Australian soldiers sang the risque song about Mrs Porter that is quoted in III: The Fire Sermon. Did Harvey know all of this or is it just a case of The Waste Land's world of echoes setting off accidental echoes of its own?

This process is apt considering The Waste Land includes allusions to show tunes, operas, folk ballads and ragtime songs among its linguistic flotsam and jetsam. In one example of cultural baton-passing Eliot took the refrain "goodnight ladies" from a 19th-century folk song and then Lou Reed took it from him.

As Radio 4's recent broadcast of The Waste Land demonstrated, it's a poem that wants to be listened to. The Fire Sermon in particular is full of noise: gramophones and mandolins, throbbing engines and pealing bells, "a clatter and a chatter". Read it aloud and before you begin to thrash out what it might mean you can hear the music humming in the wires of the verse – the "rhythmical grumbling" if you like. It's above all this sensuous, enigmatic quality that continues to inspire songwriters who want to leave all their options open; to bathe in words, and the sound of words, without locking them into a single reading. Eliot once wrote: "Genuine poetry can communicate before it is understood." True enough, many songwriters would say, especially if you can sing it.