You can't mistake Sinead O'Connor's house. Outside the porch is an empty plant pot full of cigarette butts, inside are two large statues of the virgin Mary. As the door opens, I crash into another Virgin Mary. O'Connor's housekeeper, who doubles up as her best friend, opens the door and leads me into a lounge where family photos, rocking chairs and kids' paintings jostle for pole position with more Virgin Marys. A huge beautiful bay window overlooks the sea at Bray, just outside Dublin.

When O'Connor arrives, I barely recognise her. Her hair is a black bob, her face rounded, she is wearing a three-piece suit and has the air of a mid-20th century industrialist. A big brass cross hangs down her front. "That's my ordination cross. Normally I tuck it into my bra," which she does as she speaks. She suggests we retire to the shed-cum-office in the garden. So we stroll past the hanging linen, a few guitar cases, two Yorkshire terrier puppies, the cat, and she chats away confidently, and we reach the wooden hut and shut the door on the world. Then everything changes. She sits down, just about manages to light a fag with a shaking hand and morphs into the terrified (and terrifying) wisp of a girl from yesteryear.

In 1992 she tore up a picture of Pope John Paul II on the American TV show Saturday Night live. She said it was a protest at child sex abuse in the Catholic church, and many people thought she was loopy. What abuse? Two weeks later she was booed off the stage at a Bob Dylan tribute concert, her records were publicly smashed, and that was pretty much that as a pop star.

Eighteen years on, she has been vindicated. In March, Pope Benedict XVI issued an apology to the victims of decades of sex abuse by Catholic priests in Ireland, expressing his "shame and remorse" for the "sinful and criminal acts". But O'Connor is still boiling – she regards the Vatican's admission as more cover-up than confession.

O'Connor's career is astonishing – for its brevity and longevity. It can be boiled down to the one song – a gorgeous interpretation of Prince's Nothing Compares 2 U. She looked like a skinhead angel, and sang with despairing intensity. The accompanying video was equally memorable – the close-up of that luminous face, her haunted beauty, a single tear sliding down each cheek. Nothing Compares 2 U seems to be a conventional song about lost love, but it could just as easily be about God and faith. It went to No 1 all over the world in 1990. After that there were all sorts of records – Irish folk songs, reggae fusions, self-penned compositions, but nothing much to trouble the charts. And yet she is still a source of fascination – whether for her fiery pronouncements on the church, or her unconventional approach to raising a family (four children, four different fathers), her sexuality (in 2000, she outed herself as a lesbian, then changed her mind) or her faith (in 1999, she became a priest, Mother Bernadette, having been ordained at Lourdes by the breakaway Latin Tridentine church).

Today, ahead of the start of the pope's visit to Britain on Thursday, she wants to speak about the Catholic church – not to crow, not to say I told you so, but to look forward in the way only O'Connor can. She stares into the tape recorder diffidently. "I speak very quietly," she says. She doesn't seem ready for Jesus and the pope just yet. So we talk about her new husband Steve Cooney, whom she married this summer ("Third time lucky eh?"), how he produced the first record she made, and how he was a best friend for years. And she tells me how for two years after the 2006 birth of her fourth child Yeshua, she was such a slob that all she wore was a navy T-shirt and trackpants 'til her daughter Roisin, now 14, burnt her clothes in disgust and ordered her to go shopping. "So I thought fuck it, and I love suits, and they haven't been in the fuckin' shops for years, so I got these at Next, and they're fuckin' brilliant 'cos it's like 40 quid for a jacket and 35 for a pair of trousers and you can wash them in the machine, which is fuckin' crazy, so I can wear them round the house and if the kids fuck me up it's grand."

Do any priests swear as much as she does? "Absolutely not. But that's a Dublin thing. Everybody swears. We put fuck between syllables." Part of the image revamp was growing her hair. "I grew too old and fat and ugly to get away with the bald head." She's stopped shaking.

I ask her about the Marys. She says she's always collected them. Her mother and grandmother used to buy them for her birthday. Does she think it's strange that she still has such faith? "I think there's a difference between God and religion."

As she talks I notice an inky tattoo on her elbow. "Ah, that's a conquering lion, the Rasta name for God." She rolls up her sleeve to reveal an arm that is now a series of tattooed quotes. "This is one of the names of Allah, I just got it done a week or so ago and it was incredibly painful." We work our way up her arm. "That's a quote from Muhammad Ali, who I worship – 'No Vietcong ever called me nigger'."

And this is from Psalm 91, 'So that you will not strike your foot against a stone.' "

I had always assumed that O'Connor was ordained to stick two fingers up at the Catholic hierarchy. No, she says, not at all. "I didn't do it to cause offence. It was just something private between me and the Holy Spirit." Does she practise as a priest? "I have to be very careful. I guess the way I do it is through music because people sometimes want me to do sacraments but not for the right reasons – they want me to do it because it's Sinead O'Connor."

But she says, apart from her children, her ordination is her greatest achievement. "I am proud that I did listen to that voice inside me rather than be intimidated by men telling me you can't be a priest. One ought to be more concerned in obeying what the Holy Spirit inspires you to feel rather than what a bunch of men in fucking dresses are telling you to do or not do."

I'm still trying to work out her position – she loves God, but despises Catholicism? She shakes her head. "No, what I think is wrong is that the people running the show are misrepresenting what Catholicism actually is ... what I'm talking about is the highest echelons of the Vatican't as I call it."

The Vatican't? She grins. "Yes, as in they can't admit anything, they can't stand up for anything." Where to start? Women priests, homosexuals, contraception and, of course, child sex abuse. "You can go back centuries, but the way they've behaved just in the last 20 years, over this issue of sexual abuse, shows they don't give a shit. They feel untouchable. And to me it seems they don't believe in God at all. Because if you did believe you couldn't stand in front of that spirit covering up and moving priests and doctoring reports to psychiatrists and not telling them there was a suspicion of abuse, you just couldn't do that."

She quotes any number of documents and papal decrees verbatim at me, hands me copies, insisting I doublecheck everything she says. You could imagine her in court, prosecuting the Vatican. She gives me a potted history of clerical child sex abuse – how it can be traced back to AD 320, how the first official complaint was made in 1917, the first edict was issued from the Vatican in 1922 stating that any complaints of abuse had to be silenced under pain of excommunication, how the first centre for paedophile priests was opened in 1940, how the original edict was reissued in 1962. "So they knew about it all right.

What shocks her as much as the abuse is the manner in which the Vatican claimed ignorance and suggested it is also a victim. In April, the pope's personal preacher Raniero Cantalamessa compared the attack on the Catholic church with the Nazi persecution of the Jews. "That is incendiary," she says. "Quite evil, a fucking disgrace." She is talking calmly, and stops occasionally to sip from a mug, which says: "I feel a sin coming on."

She passes me the 2009 Murphy report into the Catholic Archdiocese of Dublin, which concludes: "The commission has no doubt that clerical child sex abuse was covered up by the archdiocese of Dublin and other church authorities ... The structures and rules of the Catholic church facilitated that cover-up."

So what is the way forward? "OK, the abusive priests have been dealt with and that's very important, but now what has to be dealt with is the criminality of the cover-up." She says it has to go to the very top – after all in 2001, Pope Benedict, as Cardinal Ratzinger, issued an updated edict instructing the world's bishops to silence all abuse allegations or risk being thrown out of the church. "The Vatican is a nest of devils and a haven for criminals. It's evil, the very top of the toppermost is evil."

O'Connor is clear what has to happen – those responsible have to go. "And when all the those guys stand down we should take back the church for us." Would she like to see a democratically elected pope? "Do we need a fucking pope? Why do we need a pope? Christ doesn't need a representative. Ten years from now the church will be nothing resembling what it has been."

O'Connor's anger has always been personal. As a child, she was abused by her mother – her father was only the second man in Ireland to be granted custody of his children. O'Connor, 43, says her mother's behaviour was fashioned by a church that normalised abuse. "People under 40 don't understand what Ireland was like. It was a theocracy, like Iran, slightly less potent but the same situation. The photo of the pope I ripped up was one that had been on my mother's bedroom wall for 25 years. I took it when she died. She learned at school that violence was the way to sort her problems out. These kids were having the shit kicked out of them, and they grew up with the message that this was the way you get people to behave."

Was her mother's violence physical? "Yes, but it was also very sexual. It wasn't like she was having sex with me, but it was sexually abusive violence from when I was very small. It was horrific. I loved my mother but I was terrified of her. I literally pissed my pants if she came near me, but even when she was doing what she did to me I could see this was a soul in torment." She lights a cigarette. "I once won a prize at school for curling up into the smallest ball and the reason I could do that was because I was so used to having the shit kicked out of me." Ultimately, she says, it was the theocracy that led to her beatings that also helped her survive them. "Thank fuck I had a sense of Jesus. When I was lying on the floor having the shit kicked out of me, I'd envision Jesus on the top of some hill on the cross, and the blood would run from Jesus's heart down into mine on the floor and that's how I got through being beaten. I'd concentrate on that image."

O'Connor says she is so much calmer than she was in her 20s. A turning point came seven years ago when she was diagnosed as bipolar. "It explained a lot about being angry, fighting with people, being suicidal. And often with anger what's behind it is grief. Did you ever see this creepy cowboy movie, and at the end the guy was shot from behind and a huge hole is blown through his back – that's how I used to feel. I felt like I was walking round the world with a huge fucking hole in me. And within a day of taking the medication, I felt the cement had come and filled in the hole."

Will she always have to be on drugs? "Yes, but that is great as far as I'm concerned. Because you couldn't really live without them, you'd be in the nuthouse. Being diagnosed meant I actually had a chance of being a normal person."

The two youngest children arrive back home, jump into her arms, and tell her what they've been doing. "Someone described me as mumsy," she says, "and I love that because to me the most important thing in my whole life has been being a mum."

She's relaxed now, talking about the future. She's recording an album of her own songs but doesn't want to say much about it . "I hate talking about my career when I'm doing church stuff because it's as if I'm using the church to further my career, you know what I mean?"

Did she ever enjoy her success? "I was such an unhappy person I couldn't. The day I ripped up the picture of the pope was the best day of my life because then I became me. I could become the kind of artist I wanted to be. And now 99% of my life is rolling around the house and looking after the kids. I wouldn't go back to it for a million years."

She takes me round the house, showing favourite photos of her children. Do you think the new calm you is permanent, I ask. "People always say to me do you think your happiness is going to last, as if I'm teetering on some edge." She smiles. "Bollocks."