The Taming of the Shrew, Shakespeare's Globe Kiss Me Kate, Festival Theatre, Chichester The Only True History of Lizzie Finn, Southwark Playhouse, London

Unstoppable as a cannonball, the Bard's shrew has modern young women yelping with glee

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Beset by men of massive ego and little brain, Katherina aka Katharine bestrides both the dramatic and the musical stage in new productions of The Taming of the Shrew and Kiss Me Kate, and there is much sport to be had.

At Shakespeare's Globe, only Kate and her sister Bianca, as compliant as Kate is resistant, represent their sex in a testosterone court of servants and masters, braggarts and dolts to a man. No wonder this clear-eyed, fast-tongued captive of Padua – she should be lecturing at the city's famous university – rails night and day against the fatuous dross all around her. Young women in the audience last week yelped with glee at her poison-tipped verbal lunges and parries. Unstoppable as a cannonball, she should be playing forward for Italy, too.

Samantha Spiro manoeuvres skilfully to prepare us for Kate's final, difficult, apparently submissive speech – "I am ashamed that women are so simple ...". And by the time she advocates quiet domesticity in support of her hard-working husband, you can almost go with it, her underlying grace and rationale carrying the argument. She is not, after all, two-faced like Bianca, played with icicle clarity by Sarah MacRae, who, having sweetly nailed her man, goes her own way. And she is not delirious with self-belief or reckless with money and emotions alike.

Pissing and puking on a delighted audience as England supporter Sly, who is easily tricked into believing he is splendid, Simon Paisley Day morphs into affable suitor Petruchio, Kate's only intellectual match, who fascinates and humiliates her with his brain and bare bum. The humour in Toby Frow's production is bawdy, dished out with glee and with music, no bucket unkicked at the mention of death, no bird strutting but it be the cock or the cuckoo. Pearce Quigley's whining Grumio is wearing, but outstanding are the clowning Tom Godwin, thrumming "Go Johnny Go" on lute, and Spiro's many-layered Kate, not so much shrewish as feline.

At Chichester, Petruchio and his reluctant bride spar again in Kiss Me Kate, Cole Porter's masterly take on the Shrew. Director Trevor Nunn, with dozens of shows and lashings of Shakespeare under his belt, is at home with the ultimate backstage musical, in which the onstage warring (and its resolution) continues to the dressing room. And if you don't want to chuck in the day job to leap on the boards when you hear the curtain-raiser "Another Op'nin', Another Show", you're at the wrong theatre.

Magnificent Hannah Waddingham is the ferocious Katharine, aka actress Lilli Vanessi, Alex Bourne her Petruchio and ex-husband director, Fred, and both sing and sting with vigour and venom. But there are eye-catching performances too from spectacular dancer Holly James, and gleaming Holly Dale Spencer as a showgirl out of her depth, almost exploding in her eagerness to succeed. And above all there is melting Jason Pennycooke, who brings "Too Darn Hot" at the start of Act 2, steamily choreographed by Stephen Mear, to scorchio, knowingly doling out oodles of feelgood Factor 30.

Not everything is this well cooked: as debt-collecting goons, David Burt and Clive Rowe underplay their menace and miraculous conversion, although cuboid Rowe taking every doorway sideways is nicely done. Humour overall is thinnish – see Martin Ball in Top Hat for a masterclass in getting real laughs out of leaden old lines, a challenge that most here fail. Ace dancer Adam Garcia is sadly underused, and also stuck with the show's only duff number. A proscenium-arch theatre set wedged into Chichester's apron (design by Robert Jones) will drop neatly into the Old Vic when the show transfers to London in November.

Also doing things her way is the Irish dancer made good in Weston-super-Mare, and subject of The Only True History of Lizzie Finn, gently staged at Southwark Playhouse by Jagged Fence, Blanche McIntyre directing. Tealights in suspended jam jars over a ribbon of water stand for the stars, the sparkling sea and the theatre lighting (designer James Perkins) for Lizzie's daring, long-running can-can act. Her upper-crust new beau, at first shocked by her job, takes her back, as lady of the manor, to Kerry, where she left to escape poverty.

The Chekhovian life that awaits her, with a matriarch, an old retainer, difficult neighbours, the changing role of the aristocracy, the yearning for, not Moscow, but Cork, is acted out in dreamlike, fragmentary scenes, weakened at times by bloodless acting from Justin Avoth as Lizzie's husband, Robert. Little is explained about the new tenancy laws that impoverish the landlords, and it is hard to believe that Lizzie's savings from Weston can save the day, but Sebastian Barry writes amusingly: Lizzie (dignified Shereen Martin) thinks that poached salmon is stolen; a letter is "as formal as a hedge". And there are lessons for all, especially, perhaps, reporters with a taste for hyperbole: a freak wave is an everyday occurrence. It's just, in fact, a wave.

'The Taming of the Shrew' (020 7902 1400) to 13 Oct; Kiss Me Kate (01243 781312) to 1 Sep; 'The Only True History of Lizzie Finn' (020-7407 0234) to 21 Jul

Critic's Choice

This summer’s Almeida Festival of experimental work, in north London (to 28 Jul), includes Mass Observation by theatre group Inspector Sands and RoosevElvis, set in America’s badlands by New York’s award-winning troupe The Team. Or catch Gatz before it closes: a remarkably absorbing, avant-garde reading-going-on-dramatisation of The Great Gatsby, at the Noel Coward Theatre in the West End of London (to 15 Jul).