The opening moments sum up the show. Six schoolgirls in kilts step forward, hands folded neatly. They lift their eyes to the hills and sing a snatch of Mendelssohn like angels. But the second the strains of song die away, they slump, puff out their cheeks and emerge in the disaffected pose of youth.

And that's Our Ladies of Perpetual Succour for you – a musical adaptation by Lee Hall of Alan Warner's novel The Sopranos. Its surface is all foul-mouthed attitudes and bolshie teenagers, as its heroines use the excuse of a choir competition to go on the lash in Edinburgh. But underneath beats a heart of pure emotional gold. It is simultaneously the rudest, the most exuberant and the most touching show in the West End.

The plot is a thin thread, beautifully woven. Hall's structure and Vicky Featherstone's swift, tight direction take the girls from excited anticipation – "Let's go fucking mental" – to drunken disaster – disqualification from the choir competition when one of their number "threw up during the Vaughan Williams" – and final reckoning back in home in Oban, where all face expulsion. En route, they encounter a whole host of characters from frightened bar tenders, to prim nuns, to predatory and ridiculous men, every one of them played, with a twitch of the shoulders, or a curl of the lip and a swagger of the hips, by the six actresses who also play the choir.

The vignettes of their encounters are punctuated by music, both sacred and very profane, from Bach to the Electric Light Orchestra, all sung with zest and skill. Everything about the show gleams with care – the playing of the all-female band, Imogen Knight's choreography, Chloe Lamford's deliberately shambolic set, dominated by a statue of Our Lady with a red, beating heart. And above all the writing, which has the precision and rhythm of urban poetry.

I've seen this National Theatre of Scotland and Live Theatre co-production three times now as it has journeyed from its Traverse Theatre premiere at the Edinburgh Festival in 2015 via a national tour and a sojourn at the National Theatre which enabled it to pick up the Olivier for best comedy. Each time has been pure pleasure but what I noticed watching the opening night of its West End season is that is has simultaneously broadened and deepened. It has expanded its comedy and its musical panache – it really rocks the house – while the performances mine deeper into the subtlety of the characters portrayed.

It deserves to turn its six excellent actresses into stars – five of them have been with the production from the first, and the new girl slots in so effortlessly that she might as well have been. They find touching individuality in each of their characters who, without their care and conviction, could have lapsed into caricature.

So we grow to understand brassy Chell (Caroline Deyga) who still misses her dead father, lost at sea ("We kept his shaving things on a shelf in case he was stranded on an inlet"); reluctant virgin Orla (newcomer Isis Hainsworth, making an astonishing stage debut) who knows she is dying of cancer; posh girl Kay (Karen Fishwick) who has blighted her own life chances; sad Manda (Kirsty MacLaren) so poor yet so determined to be happy that she treats herself to "Cleopatra baths" made with two spoonfuls of powdered milk, mouthy Kylah (Frances Mayli McCann) who dreams of musical stardom; and bolshie Fionnula (Dawn Sievewright) whose horizons open and whose sense of possibility blossoms even as she sees the lives of her friends tumbled into chaos.

They effortlessly embody the play's compassionate gaze at lives that are pinioned and restricted by circumstance, but are lived by people who want to suck out of every passing moment each chance of happiness and each golden morning. It's a glorious, rude thing, full of life and hope. Everyone should see it.

Our Ladies of Perpetual Succour is currently booking at the Duke of York's Theatre until 2 September.

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