Sinéad O'Connor, Queen Elizabeth Hall

Some people – a very few – just have it. Never mind whether her songs appeal, or the style in which she performs them, but Sinéad O’Connor’s presence is extraordinary - as, of course, is her voice. She sings “I Am Stretched on Your Grave” a capella, dedicating it to PC David Rathband, the policeman blinded by Raoul Moat who recently committed suicide.

The Queen Elizabeth Hall falls to pin-drop silence; O’Connor’s singing, which flecks wrenching forcefulness with heartbreak vulnerability, is relentless - it brooks no doubt. The song itself, translated from a 200-year-old Irish gravestone elegy, is suddenly foregrounded as a thing of ancient, folkish power. It feels - it sounds corny but it really does – as if O’Connor is shouting at me alone, desperately howling her grief. Judging from the rapt faces around me, I’m not the only one with this strange sensation. You don’t get that with Jessie J.

O’Connor is often tarred with the brush of being a nutter. She is certainly an eccentric. Her outfit for this performance, for instance, consists of a black leather front-buttoned basque and black leather trousers, split by a rainbow belt, a red, gypsy-ish scarf-hat and bare feet. Combined with her tattoos, she doesn’t so much look like a pop star as someone you might see playing pool in the shadows of a seamy biker bar. She doesn’t appear to care about all the surface crap, that’s the thing, and her real mistake – if it is a mistake – has been being too candid in our tiresomely homogenised media reality. Jumping around stamping her bare feet, she is a women revelling in herself, having fun, the personal pain of recent dark times – relationship problems, bi-polar mood swings, a suicide attempt - rendered irrelevant or expelled via her art. This is as it should be for a gig that’s part of the Women of the World Festival, as is the fact that three of her band are female.

O’Connor is on fire at the moment, lit from within by a force that's mercurial but astounding

The quality threshold of O’Connor’s new album, How About I Be Me (And You Be You) is highlighted by the chunk of songs she plays from it, notably the affecting “I Had a Baby”, which lays bare a new child centring O'Connor in the face of mental illness, “Old Lady”, which she says is “about having a crush on your boyfriend’s best friend”, the joyfully funky love song “4^th and Vine” and, especially, a scorching version of John Grant’s “The Queen of Denmark” where she bites with relish into the chorus lines, “Why don’t you take it out on somebody else?/Why don’t you bore the shit out of somebody else?”

The gig’s persistent problem is the band, a big line-up replete with drummer behind Perspex sound-proofing, cellist, keyboards, backing singer, guitar, bass, and a few more too. They like to rock out and Sinéad encourages them to do so, but in truth these explosions are more fun for them than us. Partly this is because, where O’Connor’s songs have a steely edge, the band rocking out have an American FM radio vibe, like Boston or something, albeit seasoned with tribal sensibilities. Sinéad is clearly comfortable with them, and they support her in every sense, but they also drown her out. Imagine, say, Art Garfunkel singing “Bridge Over Troubled Water” submerged beneath rock guitar: that’s the manner in which it doesn’t work. Really, that voice is her instrument, so loud that one of her regular performance tics is modulating its amplification by moving her microphone far away from her mouth.

With Sinead O’Connor, less is more. She may mock her mega-hit “Nothing Compares 2 U” before she sings it but she also makes it soar, her version of new song “Reason With Me” is so raw and potent it startles the emotions, and her final solo strum through “Psalm 33”, an ode to “Jah” which she says is her favourite of her own songs, has a palpable sense of ecclesiastical intimacy. This last forms the end of her encore, after storming through “Take Off Your Shoes”, but the crowd won’t let her leave and she does a final minute-long a capella, a prayer of safety she says she learnt from some monks, and with which she wishes us a peaceful night ahead. It concludes an evening that proved to be an unexpected encounter with the mesmeric. Sinéad O’Connor is on fire at the moment, lit from within by a force that's mercurial but astounding.

Watch the video for "The Wolf Is Getting Married"