Kate Nash, Astoria, London

A year after Kate Nash came to fame via the internet, finding herself with millions of hits on her Myspace site, a mention from Lily Allen, a No 2 single and chart-topping debut album, the 20-year-old has won a Brit award as best female artist. As if that weren't enough, prior to her first tour of America, her album Made of Bricks has made the Top 40 in their Billboard 200 and she is to be the opening act at the Glastonbury festival.

For anyone who suspected she was just the pop sensation of 2007, it's become clear that things keep going up for Nash. She's retained some grassroots supporters – a sign on the balcony reads St Joan of Arc, one of her former schools. Perhaps she is the one who doesn't seem sure of how successful she's become.

Tonight, "Stop! In the Name of Love" by the Supremes, a band whom Nash says she is "obsessed" with, is playing before she enters, skipping on in a cute but demure dark blue dress, her long reddish hair bouncing over her smiling face. "Thanks so much" she gushes in response to the crowd's screams. "It's really weird playing the Astoria!" she squeals before looking around to her band for back-up.

Sitting behind the piano, she launches into "Pumpkin Soup", one of the more upbeat of her folk-pop songs, her piano playing rallying it along. But like too many of the songs that follow, she ends well-meaning pop songs with a flurry of key bashing and crescendoing vocals that end like a child's temper tantrum. It happens again – twice – in "Skeleton Song", which has a school music-lesson feel to it, with the same predictability and flippancy as a Mika hit.

But, for all the lyrics depicting teenage life – albeit a middle-class and suburban version of the Streets – it's this which connects her to her fans. "This is for those of you who have ever had a crush on someone who didn't have a crush on you", she says, before "We Get On", prompting cheers of appreciation. Her ordering the crowd to "shut up then" for a quiet number is comedic Catherine Tate.

"Nicest Thing" is a subtler song, which she plays in a trio of solos accompanied by just her guitar and which she delivers with some emotion. Her voice – when not partly drowned out by the band and her piano playing – can be exemplary, soaring and definitely the voice of one with musical training – which Nash had.

Her best moments come towards the end with her hit "Foundations" on which her voice shines, and the upcoming "Merry Happy" which can't help but allow her to be compared with Lily Allen vocals and Mike Skinner lyrics. When she introduces a song as the "limited edition vinyl single" that she "wrote a long time ago in my bedroom – I gave it to my sister and she thought I was a weirdo", you can't help thinking, or hoping, that she is going to launch into "Caroline is a Victim". This was the punchy, synthesiser-fuelled single that the indie label Moshi Moshi put out a year ago. Alas, it's B-side "Birds" that follows, another twee folk song. The crowd are delighted though, erupting into a mass singalong.

The announcement of the special guest, Billy Bragg, seems to go over most of the young fans' heads, and Nash has to repeat his name to get applause. But the duet of his "New England" is a highlight, despite it feeling more like a rehearsal. And her part in the harmonising is dubious too.

A young star she may be, but even with the next line of singer-songwriters, Adele and Duffy, to start taking the charts, Kate Nash still has her place.