The Shawl, Arcola, London<br/>Punk Rock, Lyric Hammersmith, London<br/>Katrina, Bargehouse, London

No psychic powers are needed to see that Amelia Nicholson's engrossing production deserves a West End transfer

Elizabeth McGovern is being sucked in by a charlatan. In the Arcola's excellent fringe revival of David Mamet's The Shawl – with a cast worthy of the NT – Matthew Marsh plays a cash-strapped American conman who's pretending to be a psychic. At least it seems he's hoodwinking McGovern's rich, bereaved Miss A with a caring clairvoyant act.

Amelia Nicholson's engrossing chamber production is hushed. Surrounded by the audience, McGovern sits at a black table under a glowing orange lampshade. Elegantly well-heeled, she scrutinises Marsh with a quizzical smile. He's doing the talking, lulling her suspicions with a soft intoning, his phrases rhythmically reiterated like a preacher, and with circling gestures, as if casting a spell.

This play from 1985 is short, and the plot twists – including a ruse up McGovern's sleeve – feel underdeveloped. Yet Nicholson orchestrates the jump cuts, from one scene to the next, with quiet assurance. When we see the psychic off-duty, Marsh slips in and out of his mystic persona, rationalising his art, trying to persuade his avaricious young lover (Paul Rattray) that he's simply a master of confidence tricks.

Mamet, too, is playing a clever game, exploring not just Miss A's ambivalence, but also the audience's own inclination to oscillate between smirking scepticism and a very willing suspension of disbelief. The work of a psychic medium is, indeed, very like that of a theatre-maker. While spinning yarns and feigning, he can suddenly seem electrifyingly convincing, as if truly possessed by someone else's spirit. During the séance, at the heart of the piece, Marsh is enthrallingly weird, his eyes darting nervously, his forehead beaded with sweat as he becomes a channel for multiple, fragmented voices.

The mix of the absurd and the eerie is pulled off with bravura as he breaks into Faustian Latin incantations. And you're kept uncertain, wondering if he's ultimately deceiving himself. This production deserves a West End transfer, and Nicholson is certainly a young director to keep an eye on.

Simon Stephens' new teenage drama, Punk Rock – depicting bullying, rejected love, mental breakdown and lethal reprisals in a contemporary Stockport school – is peculiarly haunted too.

One blond ghost of a schoolboy (Nicholas Banks) bounds into the closing scene, after we've seen a hair-raising atrocity committed in the sixth-formers' dusty Victorian library. The visitant vanishes as speedily as he came, just pausing to smile with gentle bemusement at his hunched, psychologically disturbed former classmate, William (Tom Sturridge). It's a touching and intriguingly elusive moment.

What's more problematic is that dramas set in schools often feel as if they're populated by the spectres of their literary forebears. It's hard not to hear repeated echoes of Alan Bennett's The History Boys, especially when Sturridge's William spiels arcane facts, trying to impress the new girl (Jessica Raine), or when he speaks of applying to Cambridge with his clever friend Chadwick (Harry McEntire).

The floppy-haired bully (Henry Lloyd-Hughes), meanwhile, is like a reincarnation of Flashman from Tom Brown's Schooldays, calling his victims "old bean". Stephens' pupil-protagonists allude to Lily Allen and text messaging, but this academy feels like a fudging of present and past. It may embrace elements from the dramatist's own boyhood and his early career as a teacher, but it's never quite convincing.

That said, Stephens is an intriguing writer and a daring one. This portrait of dangerously edgy, middle-class adolescents is often very funny as well as chilling, with quirky non-sequiturs in its dialogue alongside a tense, mounting sense of doom.

Kicking off Sean Holmes' first season as the Lyric's AD, Punk Rock also knowingly reverberates with Spring Awakening, the venue's recent storming-hit musical about frustrated teens.

And like that show, Sarah Frankcom's production offers a thrilling wealth of youthful acting talent – many making their professional stage debuts. Definitely watch out for Katie West and Sophie Wu (contrastingly gentle and snide) as well as Sturridge and Banks.

Alas, a sinking feeling is provoked by Katrina. This site-specific production, by the troupe Jericho House, sounds like an enticing epic. Staged in a derelict warehouse beside the Thames, this is a docudrama-cum-installation depicting the shocking chaos in which New Orleans was left after Hurricane Katrina and the ensuing floods.

The central verbatim-style monologues are admirably strong and politically enraging: shocking accounts of the corralling of the city's poor – with local police blocking bridges, waving guns and stealing food supplies.

What a pity the surrounding promenade is limp, in terms of both design and dramatic engagement. I'm not asking for literally "immersive theatre" but – a jazzy funeral aside – this is New Orleans with no atmosphere. You're herded through a bland tourist office to a lifeless bar where it's pre-Katrina but already feels like a ghost town. Steer clear.

'The Shawl' (020-7503 1646) to 3 Oct; 'Punk Rock' (0871 221 1729) to 26 Sep; 'Katrina' (020-7922 2922) to 26 Sep