Martha Wainwright, Regent's Park Open Air Theatre, London

Much more than just a tribute act

Hear the name Wainwright and you'll likely conjure up images of theatrical baroque pop singer Rufus, or esteemed 1970s American folk singer Loudon. Martha, younger sister of Rufus and daughter of Loudon, is less easy to recall. To her obvious chagrin, the 34-year-old is best known for "Bloody Mother Fucking Asshole", a 2005 song penned about her father.

Picking up a guitar in the magical setting of Regent's Park Open Air Theatre, her heavily kohled eyes framed by an unkempt ash-blonde fringe, she opens with "I Wish I Were" from her 2008 album, I Know You're Married But I've Got Feelings Too. Immediately, it's hard to understand why she's the lesser-known Wainwright – her huge talent could yet eclipse that of her family.

"Thank you for coming tonight," she says. "This place is spectacular!" She's not wrong. The circular open-air theatre is fringed with tall trees from which stage lights and fairy lights hang, sparkling in the darkness and casting her in an eerie light. Wainwright stands, slightly awkwardly, amid the wooden ladders and rope bridges – the set of the current production of Into the Woods – joking that if she gets really enthusiastic she might scale the biggest ladder. She kicks things off with a few well-known numbers from her self-titled debut album, warming up to the wonderfully twisted "Bleeding All Over You", from her second record – a song about a woman's scorn for her married lover who has gone back to his wife.

Tonight's performance is one of the first solo gigs Wainwright has undertaken since the death of her mother, Canadian folk singer Kate McGarrigle, from cancer in January. She pays tribute, both poignantly and with humour, to her throughout. "My brother and I have always been really intimidated by our mother's songs", she says. "But as I'm not very prolific songwriter I've started singing hers. After all, I own the publishing rights."

Halfway through her set Wainwright mumbles about "some of this Piaf stuff", referring to the French singer who, she says, "totally captivated and influenced" her. As the first bars of "Le Vieux Piano" ring out a transformation takes place. Having shed her guitar, Wainwright suddenly stands poker straight. Singing "about an old drunk woman rubbing herself up and down on a piano", Wainwright's bent arms rise up in that trademark Piaf gesture, her face changes, contorting to suit the plosive lyrics, and she hunches her shoulders to embrace the stage. She is suddenly Piaf, her energy spilling over with lusty Francophone richness.

A fabulous rendition of "L'Accordeoniste" follows. Her joyful shrieks of "La musica" produce the loudest clap of the evening. But soon Wainwright morphs back into her folk-rock persona, singing the rather depressing "This Life" and a couple of McGarrigle songs, before ending with "When the Day Is Short".

She is dragged back for an encore by a clamouring crowd and dedicates her last song, "Baby Lyrics", to Loudon. "It's good thing!" she insists, to the crowd's guffaws.