Joan As Police Woman, Classic Grand, Glasgow

This is only the second date of Joan Wasser's latest European tour, and she confesses she has "nervous bumps" about the whole experience. This gig will have taught her two things, though. The first, by her own admission, is that "before we start a show, we should make sure the amps aren't on stand-by". This was only realised around half way through, and was concurrent with a jump in volume and clarity when the magic button was pressed.

The other lesson, fortunately, was that she'll always have an unreserved welcome before a Glasgow audience. As she stifled her blush at the technical error, someone good-naturedly called out, "Start again!" There was no mad rush to claim ticket refunds, nor were there cries for the music to be turned up earlier in the set. Wasser's voice shone through and charmed the crowd very early on.

It's ironic, though, that such talent plays second fiddle to an array of more famous collaborators and contemporaries. Wasser has worked alongside Rufus Wainwright and Antony Hegarty, and – ironically – her reputation as a torch singer isn't nearly as established as either of theirs. Of her own sex, similarities may be found between Wasser and Chan Marshall/Cat Power – they both deliver a mournful falsetto which ladles on the drama – and the raw femininity of P J Harvey. She is as accomplished as neither, but her tender vocal draws attention away from any flaw that cynics might care to seek.

Those nerves are kept well hidden, as well. Wasser is, in many respects, an absolute showman – she appears in a gold-sequinned party dress, all bushy auburn hair and wide, disarming smile. It will continue to be remarked on that this is the woman whom Jeff Buckley, an icon of beauty in his own respect, was dating when he died more than a decade ago, and it doesn't look like she's aged much in the interim.

So Wasser kicks her legs up in the air to show us her gold, high-heeled cowboy boots. They were bought "on Lafayette", she informs, and she has an identical pair in electric blue. But aren't all singer-songwriters supposed to be navel-gazing and humble? This is like watching Carrie Bradshaw at open-mic night, and all the better for it.

That sense of ostentation finds its way into her music, too. Wasser has a gorgeous, smoky voice, mature but full of wonder. She uses it a bit too much, though, rolling certain notes and tones around her jaw once too often, as if the sound she's making is more important in her mind than whether we can hear her lyrics or not. The effect is to tread a fine line between dramatic and just plain drama queen.

The same technique manifests itself musically, particularly when Wasser is at the piano. On songs from her new album, To Survive, such as "To Be Loved" and "To Be Lonely" (there is a titular theme going on), she plays haltingly, her melodies replaced by smoky jazz club atmospherics. There's something about her accent and the towelled-on air of sophistication within these songs which recalls Carly Simon, although the subdued sound renders them underwhelming.

Far better is the passage where Wasser stands up with her guitar and forms something approaching a power trio with bandmates Parker Kindred (drums) and the fabulously named Rainy Orteca (bass). The singer's on-stage personality adjusts itself for this portion of the set, too – this is where the boot-flashing comes in, and where she introduces the song "Hard White Wall" with the words, "It is about fucking up against the wall".

So Joan as Piano Woman is the most earnest side of her muse, but Joan as Rock Goddess is a whole lot more fun. Such versatility means she's at least well-equipped to try to break into the premier league of singer-songwriting superstars.

Joan as Police Woman appears at the Scala, London N1 (020-7833 2022) tonight, and Concorde 2, Brighton (01273 673 311) tomorrow