The first time we meet Caroline Polachek tonight, at the left-field US pop singer’s biggest UK date yet, it is as a disembodied vocal, halfway between a wolf howl and an orgasm. A clock has counted down to her arrival, as though we were in an arena. The rust-brown stage set suggests hillocks in silhouette, the biggest implying a volcano, which, like Chekhov’s gun, is there to erupt. (No one, save the band, was expecting so much catharsis so early though.) Polachek’s three-piece are parked in one corner, as though huddled against impact.

It’s fitting that we hear Polachek before we see her. When she was 17, she sang in four choirs at once, plus an a cappella group and not one, but two nu-metal bands, on top of her opera training. As a 21st-century pop vocalist, Polachek pretty much has her own AutoTune built in (auto-AutoTune?), normally deployed with restraint. This clarion-like burst of lungpower sets the tone of her new album, Desire, I Want to Turn Into You – released on Valentine’s Day to coincide with this triumphant gig. Both mark Polachek’s move from cult nicheness to wider renown. There are many reasons why she has sold out this 5,300-capacity show; why second nights are being added to her North American tour.

Her songs are rewarding, outre pop earworms that balance defiant oddness with fashion-forward appeal. People will also know Polachek from her collaborations with artists such as Charli XCX. This is not her first rodeo, however. In 2008, when she fronted Chairlift, the band became briefly ubiquitous when their song Bruises soundtracked an iPod ad. At an assumptive glance, most people here won’t remember that first hand.

Another reason is that So Hot You’re Hurting My Feelings, from her last LP, Pang (2019), went viral on TikTok, its video’s moves seemingly tailor-made for the platform. (Polachek eventually did a TikTok of herself miming to her own dance, to widespread delight.)

As her stages have grown, so too has the music. Desire…, Polachek’s second record under her own name, is about abundance and female pleasure, hence the howl. Her Spiraling tour tilts at a kind of manic derangement, equal parts guiltless joy and nervous energy, cut through with otherworldliness: that howl, inflected differently.

The last time I saw Polachek it was in the 1,600-capacity Heaven nightclub. That was early March 2020; a few truly fashion-forward people were in masks. Polachek soon contracted the virus. Back in the US, her father also became ill and died. Polachek was locked down in London and had to say goodbye over FaceTime. It’s a fair bet that’s one more timbre in this album’s expansive vocals.

Most pop artists double as world creators now, but Polachek is one of those 360-degree operatives who has been doing her own highly granular music and art direction independently for years. Her videos are made entirely of Easter eggs; the cover for Desire… shows her on the floor of a tube carriage, performing a crawl-of-non-shame in a stained dress, with album themes scattered symbolically around. Her producer, hyperpop Briton Danny L Harle, is a passenger, his baby strapped to his chest. (He’s here tonight, as is Dua Lipa, with whom Polachek toured last year.)

Welcome to My Island is the gateway to this lush, complicated new album. And although there are obvious tropical getaway parallels with Lorde’s last record, Polachek is really asserting the primacy of her own la-la-land, while nodding to Greek myth. “Hope you like me, you ain’t leaving,” she sing-raps conspiratorially, echoing Calypso and Circe. The “heys” of the song recall everything from Taylor Swift to Simple Minds, while the sound design on Polachek’s last two records offers up a collage of dreamy pop, cutting-edge digital R&B, unexpected drum’n’bass and Spanish guitar; the album’s bagpipe soloist, Brìghde Chaimbeul, reprises her part on Blood and Butter live tonight.

Bunny Is a Rider, one of the most immediate cuts, starts as a staccato declaration of independence (“satellite can’t find her”), while Billions, a love song that coasts on thermals, unexpectedly offers something of the Cocteau Twins’s Elizabeth Fraser.

This abundance of variety has its drawbacks. Neither is here tonight, but both Grimes and Dido guest on Fly to You. Somehow, it’s a damp squib. Although drum’n’bass has been increasingly working its way into the beats of mainstream US artists, it feels tokenistic in her work, Polachek’s able live drummer notwithstanding.

Given this abundance of creativity, parallels with Kate Bush have accrued around her – comparisons she politely deflects. She’d rather be “this generation’s Caroline Polachek”, she has said. Tonight, however, it’s worth noting that Polachek is in full Babooshka mode, wearing a black leotard with gauzy chainmail skirt-flaps and tall black leather boots; she paces, postures, flounces.

Her voice cracks with emotion as she thanks Harle for helping realise her vision. And when Polachek’s voice does let rip, it is special. Exquisite high notes punctuate Parachute, a song from Pang.

Best of all, though, is her vocal on Sunset, a Spanish-themed romp about finding love. Polachek must have been peeved when the theme to season two of HBO’s The White Lotus turned up, because that track’s speeded-up digital aria is a dead ringer for her own triumphal, uncanny melisma; the zeitgeist seemingly bending around towards her again.