Sinatra: Raw (Pleasance, until 15 August) takes us inside the mind of the 20th century’s greatest crooner. The performer, Richard Shelton, catches Sinatra in confessional mode in the 1970s as he looks back on his chequered career. In the early days, a promoter suggested the stage name ‘Frankie Satin’ but his tough-minded mother, Dolly, vetoed the idea. The show’s best sections investigate the harrowing details of his tangled and doomed romance with Ava Gardener. Fame and wealth never sweetened Sinatra’s prickly character. ‘Drink is my worst enemy,’ he quips, necking whisky from a tumbler. ‘But, like the Bible says, you’ve got to love your enemies.’ This show packs a surprising emotional punch and Shelton delivers terrific versions of Sinatra’s greatest hits. No wonder the production sells out all over the world.

John (theSpace @ Surgeons Hall, until 14 August) is an Ortonesque satire that opens with a group of friends meeting at a funeral. The coffin of the dead man lies in his sitting-room. ‘It’s an open casket,’ says a mourner with a smartphone. ‘I’ll send you a picture.’ The chums are busy gossiping and drinking Prosecco when somebody notices that the coffin is empty. They set off in search of the resurrected corpse. This is a great opening gambit. How did the body vanish? Was it a real cadaver or a replica? Numerous possibilities throng the mind. But the script discards this intriguing idea and turns into a fantasy about archaeologists peddling heroin. There are some decent lines in this bungled farce but the writers need to edit and rewrite their work more diligently.

Embassy Stomp (theSpace Triplex, until 14 August) attempts to spoof the wartime spy movies of the 1950s. Quite a feeble target to choose, and the script unwittingly adopts the sexist code of the age it satirises. The best roles are taken by the three men in the cast while the lone female gets a walk-on part as a dolled-up sex object. She has the smallest role and the most expensive costume — a fur coat.

Guy Masterson provides every voice in a quickfire 65-minute version of Dylan Thomas’s Under Milk Wood. This admirable show (Assembly George Square Gardens) never quite shakes off the impression that it might be a fundraising stunt for charity.

Pip Utton has turned his amazing skills as a dramatic impersonator to a new subject, Francis Bacon (Pleasance Courtyard). Critics hailed him as ‘the greatest British painter since Turner’ and yet many of his masterpieces look like deranged and violent monstrosities. However, the man himself turns out to be a sweet-natured, old-fashioned gent with a rare talent for friendship. One-liners stream from his silver tongue. Of a sponging lover, he says: ‘He only started to read and write when I gave him a cheque-book.’ His famous South Bank Show interview with Melvyn Bragg prompts this lament: ‘Melvyn’s questions were so much more interesting than my answers.’ Bacon’s life is animated by his nihilistic philosophy, and he insists that human existence is meaningless. Art, too, is pointless. For him the only thing that matters is beauty. And though he claims that his work has no purpose or significance, he adds a teasing footnote: ‘The artist’s job is always to deepen the mystery.’ Utton’s characterisation of this cynical but hypersensitive soul is compellingly watchable. The show deserves a run at its spiritual home, Tate Modern.

Best of the Fest (Assembly George Square Gardens, until 28 August) offers a different selection of stand-ups each evening. You might catch the British-Iranian comedian, Shappi Khorsandi, a BBC and Channel 4 favourite. Her material is not especially strong and her greatest asset is a combative stage presence which, as she admits, springs from a deep need to show off in front of strangers. She exploits crass racial stereotypes but avoids censure, perhaps because the victims of her mockery are members of her own family.

Leo Kearse (Laughing Horse @ Free Sisters, until 29 August) confesses to being a ‘right-wing comedian’ and his show examines the consequences of that admission. The Scottish maestro starts by poking fun at his ‘English hippy parents’ who sold their house in London for £20,000 in the 1970s and moved north of the border ‘because they didn’t want me to be a millionaire’. He has no time for the environmental cult and its alleged cures for ecological damage. He once went snorkelling in the Great Barrier Reef where he noticed turtles struggling against a mass of disposable carrier bags and single-use Fanta bottles. The solution struck him in an instant. ‘Humans need to make plastic heavier.’ He was booked to perform at a comedy festival in Perth but the organisers revoked his contract after hearing complaints that his gags were ‘transphobic’. What they failed to realise was that he’d co-written the material with a Welsh transwoman, Dani, whom he was dating at the time. That ought to recommend him to the BBC box-tickers. But they’re probably not listening. On YouTube he’s a star.