Jerry Springer - The Opera, NT Lyttelton, London<br></br>Caligula, Donmar Warehouse, London<br></br>In Arabia, We'd All Be Kings, Hampstead, London

It's Wagner with Tourette's

The Royal Opera House and ENO ought to watch out. The National Theatre has cocked a snook by showcasing Jerry Springer – The Opera (originally developed at Battersea's Fringe hotspot, the BAC). This is a clever move by Nicholas Hytner – just installed as the NT's artistic director – for it implicitly heralds a bold new era. Besides making the South Bank look like a cheeky alternative to Covent Garden, it suggests a theatre waving goodbye to Trevor Nunn's "golden oldie" epoch with all its classic Broadway musicals.

Jerry Springer is a sung-through satire by shamelessly foul-mouthed, youthful Brits which mocks the faux-decent Mr Springer and his chat-cum-freak show where trailer-trash types confess to sexual "aberrations" before a vociferous crowd. Admittedly, this send-up of tacky TV may not wow everybody. My interest flagged now and again in the first half as the warm-up man (David Bedella) got the audience to whoop like so many trained seals, and as Jerry (look-alike cold fish, Michael Brandon) introduced various brawling low-lifers – with transvestite lovers, fantasies about pole-dancing and being spanked in diapers. This can seem like an extended sketch straining for a plot line.

Beyond that, some punters will doubtless just be shocked and appalled. Hytner's risking the wrath of the God Squad, since we descend in the second half to The Jerry Springer Show In Hell where the nappy deviant doubles as Jesus. Your more po-faced opera buffs will doubtless bridle too.

But this show is really a great musical joke. Co-written by composer Richard Thomas and comedian Stewart Lee, the lyrics are fantastically rude. Think Handel, Verdi and Wagner, all afflicted with Tourette's syndrome. The comic conjunction of the social "dregs" and "high" art is also brilliantly fleshed out by gargantuan singers who do, of course, look as if they've been stuffing themselves on Hicksville burgers.

This could be simply insulting, but there's a flip-side as Thomas's burlesque is genuinely inspired. As well as sniggering, it celebrates epic choruses and soaring arias. And thus you see how the passions played out on Springer's show may not be so far from the dramatic agonies of Rigoletto or Carmen. The score also slides seamlessly between classical, popular music and Lloyd Webberish numbers – with a surreal tap dancing routine by the Klu Klux Klan. Thomas dislodges his postmodern tongue from his cheek to allow at least one of Springer's guests a touching tune and some dignity. Elsewhere, the mix of bad-mouthing and lovely baroque harmonies make this piece as radically ambivalent as The Beggar's Opera. With divas launching into cat fights and a morally slippery (if not fuzzy) conclusion, you could indeed argue Thomas is the 21st-century's John Gay. Ultimately, this show proves more flamboyantly outré than trenchant. Lee's team is talented, slick and buoyant. Designer Julian Crouch supplies stylish projections of falling, spinning angels, Bedella's funky Satan has a stupendous (if slightly forced) vocal range, and Benjamin Lake's rhinestone-encrusted God is a divine comedy unto himself, crooning "It ain't easy being me" as he descends like a celestial barrage balloon.

The Donmar's first season under Michael Grandage continues to air modern European classics we rarely see. Albert Camus' early drama, Caligula – completed in 1940 – is a philosophically challenging and sometimes chilling study of tyranny and nihilism. What's most interesting is the Ancient Roman dictator adheres to Camus' own Absurdist view of life as meaningless, while proving that such a creed put into practice is a horror story for society at large.

Michael Sheen's febrile Caligula, who has plunged into despair after his incestuously adored sister's death, is a kind of mad, twisted purist. He raves about wanting the moon and declares he'll become a free man by pursuing absolutism absolutely. He scornfully turns the Empire's patricians into terrorised slaves and kills at random. In the end, he's half-yearning for his own gory assassination.

Grandage's darkly gleaming production – with a slate floor and gilded brick wall – feels timeless. He realises that direct visual allusions to Hitler or Saddam Hussein aren't necessary. At the same time, Sheen and his supporting cast ensure ethical arguments are invested with urgency.

David Greig's new translation brings out the black comedy of the emperor's flippancy (calling senators "duckie"). And scorching poetic descriptions of a guilt-wracked mind surface here and there. None of this quite conceals that Caligula can be a tiresomely abstruse idealist, banging on ad nauseam about making the impossible possible. There are one of two embarrassingly hammy moments as well – not least Sheen symbolically whirling in slo-mo with a mirror. But most often he's hypnotically dangerous – something like Hamlet with the thuggishness of Alex in A Clockwork Orange. He also unforgettably dresses up as Venus in drag, sarcastically demanding worship and threatening death while batting huge gold eyelashes – Sheen's Caligula certainly has an icy malignity.

Finally, Hampstead Theatre – after a slow start in its new building – presents a highly commendable premiere of In Arabia, We'd All Be Kings. This is by America's Stephen Adly Guirgis whose raw prison drama, Jesus Hopped the 'A' Train, was acclaimed at the Donmar last year. Written in 1999, In Arabia... is a darkening vision of drunks and junkies in a scruffy bar – all, metaphorically, circling the drain. Guirgis has a great ear for slang and his wasters can be irresistibly funny. This could be a grungy Cheers, at first, with Garfield Morgan as a narcoleptic old soak, Debora Weston as a scraggy slapper, and Tom Hardy as a shambling coke-head being endlessly shooed off the premises.

But Robert Delamere's production opens the play out – with a vast, ripped and soiled billboard overhead suggesting the hopelessness of those stuck in America's impoverished lower depths. That said, several story threads are underdeveloped and the downward spiral is predictable. Not a fully mature work but a writer to watch.

k.bassett@independent.co.uk

'Jerry Springer – The Opera': NT Lyttelton, London SE1 (020 7452 3000), to 30 Aug; 'Caligula': Donmar, London WC2 (020 7369 1732), to 14 June; 'In Arabia, We'd All Be Kings': Hampstead, London NW3 (020 7722 9301), to 17 May