"It's my second week at no. 1 and my album is out today," announced Lily Allen, and then giggled, which could have been a self-conscious reaction to her own bragging, or maybe she was simply tickled by the fact that things are going her way. The sweaty crowd at London's small but sophisticated Bush Hall last Monday already knew she was at the top of the charts — but cheered anyway. Even the blistering heat wave baking Britain's streets seemed timed to make Allen's summery blend of laid-back ska, pop and hip-hop seem all the more appropriate. Her single Smile is already as inescapable as the heat, wafting from open windows and workmen's radios alike.

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nginx/1.14.0 (Ubuntu) The Bush Hall performance is also 21-year-old Allen's first full-length gig, two streets away from where she grew up in west London.

Her album, Alright, Still, could possibly debut at No. 1 this week, too. And like fellow Brits Arctic Monkeys, who similarly seemed to come from nowhere, Allen's popularity was forged online. A handful of free tracks posted on Allen's page on virtual community MySpace has garnered almost 3 million listens. "Obviously I'm really grateful for MySpace and what it's done for me," says Allen, "but at the same time MySpace is just a website; it's the music that people are attracted to, not the Internet."

So what is her draw? Allen's cutting, sometimes dark, tales of London life, breakup and revenge, unwanted suitors and mortgages, drenched with wit and wrapped around the unlikeliest pop hooks and a reggae brass backing, a bit like Blondie's forays into a Caribbean sound. Allen's blog is a big hit too, as she humorously details her pop-star-on-the-rise life: "All I did was sign a deal, put up a few songs on MySpace and everyone has gotten all excited, I on the other hand am s______ myself."

Her anxiety is understandable. "I'm being followed around now by paparazzi and stuff which is really weird," Allen told Time last week. Part of her tabloid appeal is her pedigree; Allen is the daughter of Oscar-nominated film producer Alison Owen and comic and actor Keith Allen, who split when Lily was a young child. She also has a history of saying outrageous things, like telling the Observer newspaper that she sold the illegal drug ecstasy in Ibiza when she was 15. Last week, the Sun tabloid revealed tales of her time in London's Priory clinic for treatment for depression as an 18-year-old.

Allen is trying to shrug it off. "I know what the British press are like, and if I wasn't honest about my past, then it would only come back to haunt me, as it has today," says Allen. "I don't feel ashamed of anything that I've done and I don't feel like I need to hide any of it. I just think it's a bit silly, really."

But no one disputes that her life has been rocky. Allen spent much of her childhood either running away from or being kicked out of schools, more than a dozen of them. It made her wise for her age, and she delivers her reflections on everyday life in natty couplets. Allen sings over the sunny Caribbean vibe of LDN, "There was a little old lady, who was walkin' down the road, She was struggling with bags from Tesco/ There were people from the City having lunch in the park/ I believe that it's called alfresco." The little old lady then gets mugged.

Allen bemoans modern life over hyper '60s pop on Everything's Just Wonderful: "In the magazines they talk about weight loss/ If I buy those jeans I can look like Kate Moss." But the most withering put-downs are saved for ex-boyfriends — "Yeah you really must think you're great/ Let's see how you feel in a couple of weeks/ When I work my way through your mates" — Allen sings on Not Big and wiggles her little finger for anyone in the Bush Hall crowd that didn't get the subtlety of the title.

In real life she can be just as harsh. She's slammed U.S. girl group the Pussycat Dolls and former Spice Girl Victoria Beckham for "promoting womanizing" and being too thin, respectively. Not that she intends to play by anyone else's rules. "If people want to think of me as a role model, then that's up to them," she says. "I'm not going to change myself for anyone," she adds with another giggle. And as long as they keep flocking to MySpace and buying her record, why should she?

• HUGH PORTER / London
• Lily Allen is an instant hit and a tabloid target