Leonard Cohen, Opera House, Manchester<br/>Grace Jones, Royal Festival Hall, London

At 73, Leonard Cohen can still put on a hell of a show while Grace Jones, at a mere 60, menaces in a G-string

I think I've seen them all now. Not one of the pantheon of towering legends has eluded me. The ones who didn't die young, anyway. Leonard Cohen has always occupied a paradoxical place in the scheme of things: the world's most overground underground singer, the most popular of the unpopular, the most mainstream of the cult, the biggest of the little guys.

Dying young, for Leonard Cohen, was not an option: he wasn't young to begin with. Already 33 when his first album came out in 1967, to the hippie generation his role was that of a wiser elder brother played by Dustin Hoffman, teaching them harsh truths about love and sex, and providing a handbook for every bedsit beatnik and boho Romeo for whom the words "We are ugly, but we have the music" provided succour. No single male's record collection in the Seventies was complete without Cohen's little boom-tish-tish, three-strums-to-the-bar confessionals, waltzes from the edge of a candlewick bedspread. Now 73, the icon of introspection is performing in the UK for the first time in well over a decade, at a trailblazer for next year's Manchester International Festival. "Back then," he jokes, "I was 60 years old, just a kid with a crazy dream." The circumstances may not be the happiest (he's reportedly been forced back on the road after being ripped off by a manager), but those of us who get to tick him off our wish list aren't complaining.

From the start of a generous three-hour set, it's clear that Cohen's reputation as a doom-monger is misplaced. Many of his songs provide their own punchlines: "I fought against the bottle/But I had to do it drunk" is a supreme opening couplet. And moments of improvised wit are never far away. "After years of searching through the mysteries," he says, while his backing singers The Webb Sisters keep the beat, "I've finally found the key: it's 'doo-dum-dum'..."

Dressed in a dapper grey suit that looks as old as he is, he sports a fedora that doubles as a prop: often, he'll wave it like he's welcoming the troops home, or clasp it to his chest like he's burying one of them. His bony white fingers gripping the microphone, he's a frail figure now. Every time he buckles his legs at a moment of drama, you worry that his knees won't straighten again. When he introduces his band (which includes a mandolin player and church organist) about 19 times too often, you wonder whether it's a senior moment.

All three phases of his career – put crassly, seduction sonneteer, political prophet and mad monk – are represented, although, give or take the candle burning on his amp (and, arguably, "Hallelujah"), his religious side is expressed in such an indirect way that one can choose to ignore it.

Leonard Cohen can believe any kind of gobbledegook in the face of death. That's his prerogative. Just as it's our prerogative to not take it seriously, nor indulge it with any longer shrift than a secretly impatient smile.

Not that he's anyone's idea of a puritan. He stops short of performing "Chelsea Hotel No 2", denying us the line "giving me head on the unmade bed, while the limousines wait in the street", but he doesn't eschew the pleasures of the flesh entirely.

His politician premonitions, chiefly from the Eighties and early Nineties, are reaping their reward now: "I have seen the future/Brother, it is murder" is chillingly accurate, as is the grimly ironic pay-off "Democracy is coming ... to the USA", and a funked-up "First We Take Manhattan" is an exhilarating revolution fantasy.

Sometimes he almost seems to be drifting off to sleep, until he opens his eyes and squints up into the spotlights, like he's looking at an aeroplane. It's understandable. During the more meandering lounge-jazz passages, time can slow to a crawl. Other times, he's exhilarating. "So Long Marianne" still swings, and he still has that deep baritone which does strange things to women, and causes envy in men. "I was born with the gift of a golden voice", he sings knowingly, to rapturous applause.

The last we hear of Leonard Cohen is, quite perfectly, the verse "Goodnight, my darling/I hope you're satisfied/Here's a man still working for your smile."

The first we see of Grace Jones is, quite perfectly, on a screen. "Pleased to meet you," she declaims (as an overture to a brand new song called "Man Eating Machine"), her face and body mutated and distorted as if by a fairground mirror. "Pleased to have you on my plate. Your meat is sweet to me ..."

And that, in theory, ought to be enough. Grace Jones, 1980s pop star, was the creation of an interface between the worlds of high fashion and the still-new art form of video, moulded by Jean-Paul Goude with a little help from Warhol and Haring. Like Max Headroom, there were no conceptual reasons why she needed to exist in the flesh, only those of technical possibility. It tells you everything about that extraordinary cultural moment that, until Whitney came along, pop's foremost black woman was a 6ft Amazonian androgyne with a hairstyle you could balance books on.

From the moment Jones, now 60 and lured to London for Massive Attack's Meltdown Festival, appears on a hydraulic platform in a hat with a fibre-optic plume (the first of a succession of spectacular headgear that included a white witch cone, a Batman mask, Statue of Liberty spikes, a glitter bowler that shatters a green laser beam into a million shards, and a wide-brimmed hat "like my mother would have worn, being a church lady and all that"), you can't take your eyes off her. You daren't. Look what happened to Russell Harty.

Still the owner of ridiculously high cheekbones, her gazelle skeleton is emphasised by high-rise heels and a trademark shoulder-padded jacket which, once removed, exposes a corset and cheesewire-thin G-string. When she announces, mid-set, "I'm coming out naked now" – her accent teetering between comedy-posh and West Indies – it's more of a threat than a promise.

The music she made – alternating between futuristic funk, dominatrix disco and reggae's Caribbean heat cryogenically chilled to an ice sculpture – remains irresistibly compelling, whether it's those famous restylings (Iggy's "Nightclubbing", Roxy's "Love Is The Drug" and above all Chrissie's "Private Life") or the originals ("My Jamaican Guy", "Nipple To The Bottle"). And when, during her sublime bossa nova version of Piaf's "La Vie en Rose", she unleashes a full-lunged roar for the words "mon coeur qui bat", you realise what a powerful singer she is.

At the climax of a performance that's every bit as cacklingly mental as you'd anticipated, Jones invites half the audience on stage during "Pull Up To The Bumper" to shimmy and shake alongside her. They're braver than I.

Need to know

In 1994, aged 60, Leonard Cohen retreated to California's Mount Baldy Zen Centre and was ordained as a Rinzai Zen Buddhist, taking the Dharma name Jikhan ("silence"). After five years at the centre he returned to music, and has since released two albums: 'Ten New Songs' (2001) and 'Dear Heather' (2004). In 2005, Cohen alleged that, during his retreat, manager Kelley Lynch stole $5m from his retirement fund along with his publishing rights, leaving the singer with just $150,000. In 2006, Cohen won a civil suit for $9m but Lynch refused to comply and Cohen may never recoup his lost earnings.