MurleyDance is something of an oddity in the world of small independent dance companies, in that it proudly wears pointe shoes. Yes, this is – according to its own publicity - the only professional classical ballet company attending the Fringe, and Artistic Director David Murley is playing that uniqueness for all he’s worth, issuing a press release calling for more ballet companies to attend Edinburgh’s annual arts circus.

Andrew Maxwell (****) tells the Scots in the audience that he’s going to “rip the shit out of everything they hold dear” in Hubble Bubble, his take on the independence referendum. He doesn’t quite do that but it’s a witty and thoughtful take on the issues surrounding the vote.

Comics rarely start a show by referencing the ending of a previous one, but Sarah Kendall has first to do a bit of housekeeping to explain the genesis of Touchdown. The payoff for her last show was her dropping the c-bomb on her high-school gym teacher, Coach Harris, but when her mother attended a gig she said to her daughter: “It didn’t quite happen like that, though, did it?”

Young Irish actress and comic Aisling Bea made a tremendous debut with C'est la Bea at last year's Edinburgh Fringe, where she was deservedly nominated for best newcomer in the Foster's Edinburgh Comedy Awards. Now she is performing a short run at the Soho Theatre and, on second view, it's still a joyously funny show.

Massachusetts-born Bo Burnham first performed in the UK at the 2010 Edinburgh Fringe. The then teenage prodigy, who had come to fame as a YouTube sensation, took the festival by storm and was given the Edinburgh Comedy Awards' panel prize. He hasn't performed here again until this year's Fringe, when his second stage show, What, sold out in a matter of minutes and was again garlanded with rave reviews.

Tig Notaro, Gilded Balloon ****

“I've been busy. I've been growing my hair out.” Not the the most animated start to an hour of comedy, but that's how American Tig Notaro begins Boyish-Girl Interrupted, one of the most original 60 minutes I've seen at the Fringe, and certainly the most laidback.

Glenn Wool, Assembly George Square ****

There are some comics who can always be relied upon to create engaging and funny shows, and the Canadian Glenn Wool is one of them. His comedy appears to be straightforward stand-up – anecdotes are interspersed with one-liners and puns, with occasional interaction with the audience, to create a small world of his own, with more than a touch of the surreal about it.

John Lloyd, Underbelly Bristo Square ****

Carey Marx, Gilded Balloon ****

Carey Marx couldn't come to the Fringe last year, because of the small matter of having a heart attack. But, looking on the bright side, the experience has given him his new show, Intensive Carey, in which the comic tells his story without a trace of self-pity and with a keen sense of the absurd.

Gyles Brandreth, Pleasance Courtyard ***

This is an agreeable hour of theatre and political anecdotes that former MP and now BBC presenter Gyles Brandreth tells with great aplomb. He drops a lot of names, but he's very good mimic – John Gielgud, Frank Sinatra, Prince Charles and others make an appearance – and the stories (whether wholly true or not) are very funny.