Frank Zappa once said that most rock journalism involves "people who can't write, interviewing people who can't talk, for people who can't read". I think I begin to see what he meant after spending an entertaining hour with MIA. The singer is perfectly beguiling, but quite baffling, and I leave wondering what on earth she has been talking about, how on earth I am going to write it up, and whether any reader will ever manage to make sense of it.

To give you an example, here she is talking about her Sri Lankan identity: "Well in the beginning no one knew where the fuck it was, so I never really talked about it. And especially to explain: 'Well, yeah, and there's the Sinhalese and the Tamils and the war. You know, by the time you got to England you were just brown. So it was like going through the sort of filtration process. And to take it back to such a minutiae thing of going, 'And then there was like this and da da da da da' – so once you're just operating on being a Paki, you know, then it wasn't about that, it was just, 'Oh well, that's obviously wrong and you're not going to go deep into that thing', and for me life was just about, you know, connecting with people through music."

Much of MIA's conversation is not unlike a work of abstract art; once transcribed on to the page, it can be read and re-read and parsed for meanings, of which any number might emerge. Her sentences frequently end with, "Do you know what I mean?", but I very seldom do, so I tend to have a stab and suggest an interpretation, with which she usually concurs. But whether it really was what she meant is anyone's guess, for I get the feeling that were I to propose an entirely different meaning, she'd just as easily agree with that one instead.

One thing, at least, does become clear. MIA's biography has always been fascinating, but notoriously hazy, prompting scepticism in some quarters about her authenticity. But having now met her, I doubt the confusion derives from any deliberate fabrication on her part. Instead, she just doesn't really talk in a register conducive to the banality of facts and clarity.

A summary of her life, as far as we can tell, goes like this. Born Maya Arulpragasam in 1975, in London to Sri Lankan Tamil parents, she was two months old when her father popped out "to buy a pint of milk", and disappeared off to Lebanon to train with the PLO. When he returned four months later he announced that the family was returning to Sri Lanka, where he founded a Tamil student protest movement called Eros. Her father was almost entirely absent during her childhood, but the government believed he was close to the Tamil Tigers and so soldiers would raid her village, demanding to know where he was and beating up her mother and other relatives. By 1986 the situation had became too dangerous and the family moved back to south London, without him.

As a teenager in London she hung out with West Indian kids, falling in love with dancehall and hip-hop and fashion. "That's very rare for Sri Lankans, cos we all get taught to be doctors, so the Sri Lankans were like: 'That girl there, do not hang out with her, she's terrible.'" After A-levels she went off to visit a cousin in Los Angeles and lived as a "ghetto queen" among the gangsters and rappers of Compton, before enrolling at St Martins in London to study fine art, where she met Justine Frischmann, lead singer of Britpop group Elastica, who hired her to film the band on tour, and design its album cover. In due course she began playing around with Frischmann's old synthesiser, came up with the stage name MIA, and in 2004 released her own hit debut album, Arular, which was nominated for the Mercury prize and hailed by the rapper Nas as "the sound of the future".

It was a cleverly mongrel sound – inventive and infectious, part hip-hop, part grime, shot through with snatches of world music, but underpinned with an emphatically street aesthetic – making MIA wildly cool, if not hugely commercially successful. A reputation for edgy dissidence was reinforced by her public pronouncements on politics – chiefly, though not exclusively, about the plight of the Tamils in Sri Lanka. But her second album, Kala, released in 2007, featured the hit single Paper Planes, which was nominated for a Grammy and remixed for the soundtrack of Slumdog Millionaire, pushing MIA from critical acclaim to mainstream stardom.

MIA at the Grammy awards in February 2009. Photograph: Robyn Beck/AFP/Getty Images

Admirers began to talk of her as a modern Madonna – as much an inspired editor as a musician, with a genius for poaching from the worlds of fashion, art and music to create something accessible and contemporary. Last year she got engaged to Ben Bronfman, heir to the Seagram fortune and son of Warner's CEO Edgar Bronfman, moved to an upscale house in LA, performed alongside Jay-Z and Kanye West at the Grammy awards, and gave birth to a baby boy three days later. Since then she has been in the studio in LA, working on a third album widely expected to consolidate her place in pop's stratosphere.

The album, /\/\ /\ Y /\ or Maya, was released this summer, and some of it is, well, it is what critics like to call "challenging", and a layperson might describe as sounding like jumbled-up industrial roadworks. My one-year-old son will dance to just about anything – even the Sky News theme tune – and when I played Maya to him he did his best, but even he soon had to concede defeat.

It's a defiantly aggressive piece of work, and yet in person the singer is petite, languid and strikingly demure, with an almost dainty bearing and a soft, sultry drawl. She is quietly watchful, rather feline, and nothing at all like the brash, gobby presence of her live performances. But at 35, MIA has been working all her life towards the fame and success her second album seemed to promise – so why did she choose to reject it by making such a difficult album now?

"Well with so many people wanting that thing," – mainstream fame – "people forget the idea of not wanting it. So it was a huge shock for everyone, including my label. I think maybe it's just that people forgot what it felt like to be punk, and punk's not this new thing that's been adopted by LA or Hoxton people with spiky hair, it was genuinely about people who didn't want it, who just saw things differently. So yeah, it was weird because people see it as a failure as opposed to something that celebrates art or champions art. People go, 'You fucked up.' But no, it's just, it's not that I thought about it, it's just how I felt. It's way more important for me to be creative and push boundaries in myself and for myself, cos it's part of my evolution and growth as an artist, you know. I needed to go here to this particular place because my personal life is that contradictory and weird and insane, and it was trying to reflect that."

Her lawyer was quoted earlier this year saying that MIA "vacillates between wanting to be huge and maintaining her artistic integrity. That's her dilemma. Because if you want to be huge you have to give up a lot." I ask if this album represents her resolution of the dilemma.

"Well it's like this. At the time when I was making the record this civil war [in Sri Lanka] came to an end, you watched all these people die on the internet, no one did shit about it, the government got away with it and they made loads of money on top of that, they got given billions of dollars on top, you know. And you can see that they're obviously really bad. If I wanted to be really rich, that's what I'd do. I'd call myself a fucking government, kill a load of people, and you get billions of dollars, you know that's the easiest way of doing it."

You don't need to spend long with MIA before Sri Lankan politics come up. She has been an outspoken advocate of the Tamil people's rights, and tireless in her determination to draw attention to the country's largely ignored civil war. As a consequence, she says, she has been branded a terrorist sympathiser by the Sri Lankan government, whose agents have contacted her fans, threatening them with prosecution under terrorism legislation if they post her music videos on the internet. She was also denied a visa to the US while making her second album, which she puts down to her political profile.

The frustration is that while everything MIA claims may very well be true, she doesn't always couch it in the most reliable of terms. For example, she has said that within months of moving to LA her house was bugged and her phone tapped, but when I ask about the evidence for this, she says:"Well my mum's still not allowed into the States. I've hired like six lawyers, and the Bronfmans have tried to help. And the only thing I've heard back is that maybe she spoke to someone she shouldn't have spoken to. Which makes me think, well, her calls are logged."

She frequently asserts that Google is funded by the CIA, but when I wonder if this might suggest an element of paranoia in her thinking, she offers casually: "Well I don't know, how do you know that it isn't funded by the CIA? How do you know that the hedgefunders didn't come ultimately with money that comes from there?"

By that token, I laugh, how does she know I'm not a CIA spy? "Well that's my, like, two years in America, where I have found out these things, cos people I sit next to at dinner parties do speak like that. Now I'm never going to be invited to dinner again." What does she mean? "Er," she offers coyly, and pretends her lips are sealed.

MIA's political statements are a fiery antidote to the anodyne culture of contemporary pop culture, but critics have dismissed them as too subjectively emotional to be taken seriously. Does she see any truth in the charge that they lack analytical weight?

"Yeah, well I would've been a politician if I wanted to discuss politics in that way, but I, yeah, I like to stick at being, you know, a person who speaks about the experiences that I experienced, like I have the right to. All I can do is when I come across people going, 'You know, in order to make music you have to be like this,' I go 'No, in order to make music you don't have to be like anything, you just have to be musical.'

"And if you happen to have this other experience it's OK, because my point is that as long as there's wars there's always going to be refugees. And as long as there's refugees there's always going to be some kid going, 'I'm shit.' And you're going to be like, 'No, you can just be whatever you fucking want to be.' And that's how I started. It's not that I bang on about: 'Hey I want to adopt white children from Hollywood, or go to Scotland and save babies, and save Aids, and, you know, Africa, and save, like, the starving people.'"

Yet she does acknowledge a certain, if not unease, then ambivalence about her shifting identity from refugee street artist to LA aristocracy. She gave birth to her son in one of America's finest hospitals, "and it cost $48,000, and it was weird like going in to have a child knowing that, you know, most of the people can't afford it." She has always, she says, wrestled with the tension between her multiple identities, ever since St Martins, when relatives were losing their lives in the Sri Lankan civil war while she was trying to make sense of the decadence of "white middle-class males" and their arty preoccupation with apathy. And now she is marrying into millions.

MIA at the Scream awards in October 2010. Photograph: Matt Sayles/AP

"My record label always says you shouldn't talk about money because it makes people extremely uncomfortable. Refugees can't talk about money. Rappers can talk about money, refugees can't talk about money. And I'm like, what the fuck? No. There's no rules to any of it. Cos yeah, this is an issue, it's a fucking dilemma, you know, I fell in love with someone who's got money." But then she adds, "I don't have that much money. Just cos you marry your fiance it doesn't mean you get your father-in-law's credit card."

I can sense that she's trying to answer a specific criticism – but I'm not sure which one. Is she, I ask, alluding to a New York Times profile earlier this year, which seemed to imply that she was living the high life while still pretending to speak from the street? At the time she was so angry about it that she published the journalist's mobile phone number on Twitter.

"No," she laughs. "I was talking about my ex-boyfriend saying that because you're exposed to your fiance's family, having loads of money, it makes you no longer care about your art. And I think that's bullshit. Cos this record was way harder for me, and it's way more personal, and I've put so much more blood and tears into it than anything else I've ever done. And it was really difficult, cos I knew every minute I spent on that record it was dividing the time I spent with my baby and on my art. And that to me is the ultimate, you know . . ."

The ultimate what? "The ultimate, like . . . The ultimate . . . I don't know. Do you know what I mean?"

We're drifting deeper into a fog of vague allusion when, suddenly, she seems to focus, and out of nowhere comes perhaps the most perfect exposition of her creative credibility.

"I don't know why it's not a celebratory thing, the fact that I just know about a lot of fucking shit. That's all. Yeah so I know how billionaires live in America, and I know how poor people live in Sri Lanka, and I know how soldiers are, and I know what it feels like for your dad to throw hand grenades out of your bedroom window, I just know that. I'm not going to be able to change any of those things, and ultimately I believe in creativity. You get out what you put in, and it's not like I only put one thing in."