Double Falsehood, Union Theatre, London<br/>Becky Shaw, Almeida Theatre, London<br/>Little Platoons, Bush Theatre, London

A few passages in this 'new Shakespeare' sound suspiciously 18th-century, but more often its flow is credibly Jacobean

This is pretty sensational news on the theatre scene: a long-lost play by the Bard, startlingly added to Methuen's canonical Arden Shakespeare series and now receiving its first professional staging in more than two centuries. 'Ods bodkins!

What's really fascinating about Double Falsehood, however, is its hotly disputed, complex provenance. As the Arden edition itself acknowledges, the authorship question remains vexed regarding this tragical-comical-pastoral romance wherein two young lovers are unhappily separated; a rape victim seeks recompense, disguised as a shepherd-boy; and a two-faced knave called Henrique is eventually exposed.

It was first presented as a great rediscovery at Drury Lane in 1727, having apparently been unearthed by Lewis Theobald, an ambitious writer and Shakespeare editor. However, it was then dissed as a forgery, a drama not only depicting a duplicitous rotter, but also concocted by one.

Theobald was trounced in Pope's satirical Dunciad. Today, however, a growing number of scholars believe he genuinely acquired – and only tinkered with – a script of a drama co-written circa 1613 by the semi-retired Shakespeare with John Fletcher (originally entitled Cardenio).

For theatregoers off to see Phil Willmott's production at Southwark's pioneering Union Theatre – a scruffy burrow under a railway bridge – the fun lies in trying to determine which bits, if any, were penned by the master. It's Spot-the-Bard.

There are, for sure, a few passages that sound suspiciously 18th-century, stiff and moralising. But more often the poetic imagery and pentameters flow by, credibly Jacobean. They're spoken with winning verve by Gabriel Vick's ardent, faithful Julio and his sweetheart, Emily Plumtree's Leonora who, when parentally pressurised to marry his treacherous friend, is ready to commit suicide on the wedding day.

Though not scintillating, Adam Redmore's Henrique has one fascinating flash of guilt, sloughed like a snake cast as he moves on from assaulting the maid, Violante. Meanwhile, Julio's father (Stephen Boswell) is a delight: frail yet explosively fuming at snobs.

Played out in modern dress on a near-bare stage, Willmott's production begins lucidly (and making Leonora's parent a matriarch, rather than a patriarch, works OK). But then monks' cowls are introduced, confusingly doubling as shepherds' smocks. As for the script, if Shakespeare was contributing, he wasn't firing on all cylinders. What might be trademark Shakespearean narrative ingredients and figures of speech seem increasingly like generic regurgitations, with a jolting plot (surely missing scenes?) and echoes of everything from Romeo and Juliet, As You Like It and Measure for Measure to Hamlet and Lear when Julio goes mad.

Still, Double Falsehood is an intriguing curio, and I greatly look forward to seeing it reworked in Gregory Doran's imminent RSC staging (entitled Cardenio).

In Becky Shaw, the Almeida's new domestic drama by the US writer Gina Gionfriddo (of TV's Law & Order), Suzanna and her adopted brother Max are trying to shake off their semi-incestuous hankering for each other. This may sound scarily screwed-up, yet Gionfriddo surprisingly attaches it to a dry, witty, upmarket sitcom.

Peter DuBois' production, importing David Wilson Barnes from off-Broadway, is highly entertaining and extremely slick, with hotel rooms, apartments and cafés on a revolving stage.

Max is a wealthy financier with no time for social niceties or self-pitying psycho-babble. And every line that Barnes delivers is scorchingly funny as he gives Suzanna's soppy liberal husband, Andrew, short shrift and then gets cornered by a madly grinning blind date. Daisy Haggard is hilarious as Becky, bug-eyed but not as dumb as she seems. Anna Madeley's and Barnes's suppressed desires are startling too.

Nonetheless, Vincent Montuel is a mild bore as Andrew, and Haydn Gwynne slightly cheesy, having to play the rich, sourpuss mother then serve as a mouthpiece for wise saws. In the end, the writing feels terrifically smart but slightly smug: a bit like Frasier with a nod to the darker terrain of Sam Shepard.

Lastly, in the Bush Theatre's Schools Season, Little Platoons by Steve Waters is more of a letdown, oddly scrappy and half-baked. It sounds terrifically topical, being based patently on the "free school" currently being set up by Toby Young and his west-London buddies. And Waters promises to show how selfish motivations can undermine Big Society initiatives.

Nathan Curry's production sits on the fence, though. It wavers towards but then steers away from a trenchant, barbed tone. Only Joanne Froggatt is distinctly satirical as the perky government bureaucrat, Polly. Andrew Woodall does his best to give Nick, the swanky maverick spearheading the new school, both engaging charisma and a bullish edge.

Meanwhile, Claire Price adopts an irritatingly worthy manner as his hastily corralled headmistress, who's embroiled in a child-custody battle. There's also a cameo appearance by four comprehensive pupils with swagger, but they're basically just lifted from the other play in the season. B-minus. Could do better.

'Double Falsehood' (020-7261 9876) to 12 Feb; 'Becky Shaw' (020-7359 4404) to 5 Mar; 'Little Platoons' (020-8743 5050) to 29 Feb

Next Week:

Kate Bassett reports on The Children's Hour, Lillian Hellman's 1934 school drama, starring Keira Knightley

Theatre Choice

Bruce Norris's sharp satire about race relations, Clybourne Park transfers from the Royal Court to Wyndham's (to 7 May). Comic yarn-spinner Daniel Kitson revives The Interminable Suicide of Gregory Church, a curiously comforting saga about an attic full of morose letters: tour starts at Liverpool's Everyman (Wed to Fri), then West Yorkshire Playhouse (Sat).