When it comes to spotting stars in Los Angeles, there’s a phrase that’s popular among the paparazzi. It goes, “If you think it’s somebody, then it probably is.” When Lily Allen arrives at a beachside restaurant in Santa Monica, she is palpably “somebody”. Sporting a knee-length Chanel coat, a Fifties cocktail frock and her trademark trove of conspicuous gold jewellery, she’s flanked by a manager and an assistant. Marching to a table, she sits herself down and orders a Bloody Mary and a bowl of chips. Though she’s not actually done much, the scene draws glances… then second glances, as staff and patrons all rack their brains trying to work out precisely who she is. Everyone seems a little confused that they can’t quite place her, so they go back to serving tall glasses of carrot juice or nibbling at salads.

“I’m bored of not being famous here,” Allen says, rolling her eyes then grinning. She’s certainly bored of jumping through the promotional hoops that generally precede stardom in America. On the home stretch of a whistle-stop five-city tour, she describes the relentless itinerary pushed by her US label. For someone who merrily describes herself as being “f****** lazy” and who “can’t stand being told what to do”, she’s holding up. But only just.

“At least in New York there was a British paparazzo following me for a while,” she smirks. She describes how she told her chauffeur to wait as the “pap” frantically hailed a cab, and then allowed him to tail her tantalisingly for a dozen blocks before she vanished into the traffic. It’s about all a girl can do for kicks right now. She asks if she can smoke, but it’s a forlorn hope in California, even at an outside restaurant. Muttering about it being “enough to make you give up”, she phones her mum (film producer Alison Owen, whose work includes Shaun of the Dead and Elizabeth, and with whom she still lives in Islington), moans that she never hears from her dad (actor/comedian/pug-faced lothario Keith Allen), before remembering that, actually, she spoke to him the previous day. Then suddenly she’s gone, dashing off with her manager and assistant to a meeting. She’d managed to take one sip of her Bloody Mary.

It’s hard to imagine the Lily Allen of Blighty being at the beck and call of anyone. Coming from just about nowhere at the start of the year, by July the 21-year-old had breezed to No 1 in the charts with her self-penned debut single Smile, a clement cut of reggae-pop that belied the rancorous venom of the lyrics (When you first left me/ I was wanting more/but you were f****** that girl next door/what you do that for? the song opens).

Proving she was no summertime flash in the pan, her second single LDN (a text-speak abbreviation for London) went top five, itself an airy ode to the capital, replete with references to muggers, pimps and “crack whores”. The release of Alright, Still, her album, only confirmed what many had already begun to suspect – that here was that rarest of things, a proper pop star; young, talented and interesting, someone who could convince you that Stock, Aitken, Waterman had never really happened.

Today her appeal is so broad that she’ll happily explain that the biggest problem her label faced during promotion was trying to work out where to place the TV ads for her album; during programmes for kids who might otherwise be spending pocket money on Girls Aloud? During student TV for twentysomethings not wanting to miss a hip new trick? Or during the evening news, for adults who might recognise her debt to Althea and Donna or the Specials and want to find out more? In the end, the label did all three.

Songs aside, it’s a lack of tits-and-teeth media grooming that’s accounted for much of Allen’s popularity. And while the music and fashion press fell at her ice-white, box-fresh Nikes, the tabloids must have viewed her arrival with glee, a steady stream of honest, if opinionated, quotes on drugs, sex and the relative merits of other celebrities proving endlessly malleable. That she’s Keith Allen’s daughter was the icing on the red-top cake, and she was quickly labelled a “Pugnacious Pop Pixie” and “Loudmouth Star”. This need to caricature Lily as some gobby good-time girl sort of makes sense. It might not necessarily be true, but it is a neat way of getting a handle on someone whose background and persona seem relentlessly contrary.

There are the superficial contradictions, such as her being a white girl making reggae and hip-hop-influenced music. There’s the fact she’s attended fee-paying schools such as Prince Charles’s old prep school Hill House and the famously progressive Bedales, but still maintains an air of tower-block glamour. She’s charming and warm, but makes no bones about describing someone like Bob Geldof in crudest four-letter Saxon on her MySpace site. An oft-stated desire for comfy domesticity rather than chart domination has led armchair psychiatrists everywhere to point to her dearth of family stability (Keith Allen left Lily’s mother when Lily was three, and has recently downplayed the importance of fathers in raising kids), plus her 13 schools, as the latent cause. For all the media coverage, there are still a lot of loose ends to Lily.

We reconvene that evening in the garden bar of her hotel, the Chateau Marmont on Sunset Boulevard. Referred to as “The Chateau” by the exclusive clientele who can afford to stay here, it has a healthy rock’n’roll pedigree. It’s where John Belushi died. Led Zeppelin once rode motorcycles through the lobby. Jim Morrison fell out of one of its windows. Guests aren’t encouraged to emulate this sort of behaviour, which is just as well, as by now Lily is knackered and slumped in a cosy armchair. The only concession to wild abandon is that she’s allowed to smoke, and she warmly thanks the waiter who has brought her a huge hot chocolate and a packet of cigarettes.

I ask how much she smokes. “It depends how boring the interview is,” she snaps back. “Do you smoke? It’s very cool, you know,” she adds sarcastically. She taps ash into the ashtray, tucks her knees into her chest and mumbles. “Right now, I just want to climb into bed with my boyfriend. Not that I’m moaning,” she continues, in the tentative tone of someone about to moan. “But I didn’t think it would be this hard. I never realised how much of this job is just selling yourself as a product.”

And as the words trail off, she squeezes her eyes together and breaks into short, sharp snorts of laughter. It’s as infectious as it is unexpected. “That’s the arrogant thing about me,” she recovers. “And it’s why I never succeeded at school… I just hate being told what to do. I’m an incredibly stubborn person, and I don’t want to get out of bed for anyone”. She makes good her claim the following day, giving herself the day off and cancelling an interview with a national American magazine.

Now the minor moan is off her chest, she perks up considerably. At one point she nearly keels over with excitement when Lindsay Lohan plus entourage stride past (“She lives in this hotel… God, I really want to be her friend,” she squeals. “Maybe I’ll send her a note”). Further buoyed by this encounter, she recounts the previous week’s surprise invitation to the Chanel fashion show in Paris.

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