As Mae West almost asked, is that a pun in your pocket or are you pleased to see me? Certainly the answer offered by this delightful spoof of 1940s Hollywood musicals is yes, and yes again. Puns, quips, doubles-entendres, malapropisms and jolly repartee seem to come pouring out of every part of the stage, from the palm trees to (yes) the characters' pockets; and the feeling is so ebulliently welcoming we found ourselves helplessly chortling at what we might have sniffily dismissed as Christmas-cracker silliness.

The programme says that Dick Vosburgh and Denis King have based their show (at the Apollo) on Moliere's Imaginary Cuckold, a play unfamiliar to me and, I suspect, to Moliere himself. Still, the alleged debt allows Andrews Sister lookalikes to bounce on, singing that if the great Frenchman had gone to Hollywood "I know he'd be raking in the dough". That blend of earthy fun and sly sophistication typifies what follows. Imagine Cole Porter contributing to a collaboration between Groucho Marx and lrving Berlin, and you have some of the rhymes and much of the feel.

My expectations, I admit, were less high. Perky musicals that succeed in a friendly pub-theatre - and this comes from the King's Head - can look a bit tinpot in the West End. Moreover, I wasn't hugely taken with the idea of bringing on characters evoking West, W.C. Fields, Jimmy Durante, Abbott and Costello, Rita Hayworth and Gene Kelly to perform a sentimental romp about sailors in town. It sounded nostalgic and irritatingly knowing. It sounded a pain.

Well, nostalgic it is, but a pain it isn't, thanks to Ned Sherrin's refusal to let his production get excessively self-parodying, to King's period hums, and, above all, to Vosburgh's unstoppable words. The plot is predictably preposterous. Barry Cryer, alias Snaveley T. Bogle, alias a squinting, saturnine Fields, is married to Pauline Daniels, alias a Mae West who majestically wiggles about in dresses that make her look like a lacy Boadicea or a vast rococo raspberry. Gavin Lee's tap-dancing Danny decides Fields has commandeered his fiancée, Rae Baker's gorgeous, flame-maned Anna. West draws some wrong conclusions, too - and so to a denouement as flimsy as tinted celluloid.

But this matters not at all when Brian Greene's confused Durance is declaring himself on the horns of a Dally Lama, or Vincent Marcello and Michael Roberts's Abbot and Castle launch into yet another wonderfully goofy routine, or someone is accusing Fields of being two-faced and someone else answering that "if he had two faces, why would he be wearing that one?" Nowadays you are not meant to crack jokes about people's looks, still less their alcoholism or nymphomania; but when the show has a go at sex-mad West or at tipsy Fields, I found it wickedly refreshing.

She sings a splendidly robust song about Danny being the frankfurter in her bun, the banana in her pie, the organ in her chapel, and invites him to her room for a late breakfast consisting of "just a roll with a little honey". He cracks lugubrious jokes about the evils of the demons water and milk, threatens to annihilate an enemy by breathing on him, and attributes his permanent hangover to "getting a bad piece of ice". Did we laugh? You bet we did.

Those of a serious disposition, who view a visit to the theatre as the opportunity for an improving experience, should read no further. A Saint She Ain't is the silliest, funniest musical in recent memory, crammed with more jokes than seems decently, and in many cases indecently, possible. The book and lyrics are by Dick Vosburgh, an American in London who has clearly misspent years of his life curled up in the sofa watching old Hollywood movies on the telly. His diligent research now bears marvellous fruit. The show, first seen at the King's Head in north London in April and now deservedly transferred to the West End, is a glorious send-up of Hollywood musical comedies of the forties, the kind of picture that, regrettably, they don't make any more. And, with a nice theatrical twist, the daft and insubstantial plot is based on a rarely performed Moliere play,  Le Cocu Imaginaire.

The fleet is in town in the Hollywood of 1944, and our adorable Rita Hayworth-like heroine, Anna, has fallen head-over-heels in love with Danny, a tap-dancing sailor in the Gene Kelly mould. But in this kind of caper the course of  love never runs smooth. Anna's Jimmy Durante-like dad wants her to marry a rich heir; worse still, due to an absurd mix-up involving a locket, Danny the sailor thinks that Anna has become involved with a WC Fields-like drunk, while Anna suspects Danny of having an affair with the drunk's wife, a voluptuously predatory Mae West-style vamp. "So long," says Danny, attempting to beat a retreat from her advances. "Mmm," she drawls in reply, eyeing his crotch, "let me be the judge of that. "

It isn't all perfect. A confirmed Hollywood nut I met in the interval complained that some of the impersonations were wide of the mark, and Ned Sherrin's production could sometimes do with a touch more oomph. But there is an infectious feeling of shared pleasure coming from the stage which I found irresistible. Denis King's score offers clever showbiz pastiche, though only one tune has lodged itself in the memory, a staggeringly dirty ditty of outrageous double entendres, delivered by Pauline Daniels's wondrous Mae West and called The Banana for My Pie. (The symbolism is every bit as obvious as it sounds and there's more, much more, in similar vein.)

Barry Cryer is in terrific form too as the WC Fields figure, balefully delivering deadpan jokes about the booze, while revealing a strangely touching affection for the vampy wife he fears has cuckolded him. Brian Greene is the lucky recipient of a wonderful parade of malapropisms in the Jimmy Durante role - "Boy am I on the horns of a Dalai Lama," he announces at one point - while the tall and achingly beautiful Rae Baker, so good in Chichester's Nymph Errant this summer, once again proves that she is a musical star in the making as the flame-haired heroine. There's strong support from Michael Roberts and Vincent Marzello, who have some inspired cross-talk routines in the style of Abbott and Costello, while Corinna Powlesland is a delight as Anna's toothy best friend, who is desperate to get a man into matrimony. The V on her sweater stands for virginity, she announces, but it's a very old sweater.

By now you will know whether this is the show for you. Many will undoubtedly find it unforgivably puerile. Those, like me, who believe that a dirty mind is a joy for ever, will have a ball.