Amanda Palmer, Village Underground, London Tom Petty and the Heartbreakers, Royal Albert Hall, London

Amanda Palmer's fans, who funded her album, are rewarded with an unforgettable show

You're more likely to read about Amanda Palmer in the pages of The Economist than in the NME, a fact which highlights just one reason why Palmer is one of the most extraordinary artists in the world of independent music.

The Bostonian singer – one half, with drummer Brian Viglione, of The Dresden Dolls – recently made headlines outside the music press by pursuing an alternate business model with enormous success. The practice of "crowdfunding" is nothing new, but Palmer has taken the principle to new heights.

Always close to her audience, Palmer used her Kickstarter page to raise a staggering $1.2m to cover the costs of her forthcoming album Theatre Is Evil, for which she thanked them by baring her breasts in a photo. If only they'd known, they could have seen them in the flesh for free.

One reason for the devotion of her following is her ability to unlock the creativity in others, whether in music or other disciplines: take, for example, the thriving Fan Art section of her website. That creativity is on show at Village Underground, a bare-brick warehouse-cum-art house in Shoreditch. Amanda has decorated the space with paintings of, or inspired by herself, courtesy of artists as diverse as Robyn Hitchcock, DJ Spooky and Palmer's own husband, the comics writer Neil Gaiman.

The room falls dark, a fire door swings open and, in a Lone Ranger mask and cream silk ballgown, Palmer enters, led by a bagpiper (her cousin Hugh), clutching a bizarre array of instruments and props and bidding us to sit cross-legged around her.

For the first song of a show which is unamplified, she keeps time by rhythmically whetting a kitchen knife. For the second, "Trout Heart Replica", she neurotically whittles a root vegetable on a chopping board. Never has a beetroot been sliced with such quiet tragedy: it's like the Morecambe and Wise breakfast sketch directed by Ingmar Bergman.

That song's animals-as-metaphors-for-trapped-humans theme continues with Nirvana's budgerigar-based "Polly", one of a number of covers: we also get Radiohead's "Idioteque" and Lou Reed's "Walk on the Wild Side", during which she teases our British reserve by deconstructing "the way the Germans know all the words and Londoners don't".

Using white sheets and a pillow, she acts out "The Bed Song", with the astonishingly bleak couplet "You take the heart failure, I'll take the cancer". It's a mistressclass in the power of DIY and imagination. Gaiman himself pops up for a charming recital of country legend Leon Payne's "Psycho", a serial killer's confession which is what "Bohemian Rhapsody" would be like if it was written by Tom Lehrer.

Palmer ends with the "Ukulele Anthem", which advances the argument that the humble "piece of wood and plastic" could end youth crime. Suddenly, she takes off all her clothes and, standing naked, invites the crowd to scribble all over her, a clever reversal of the usual autograph scenario.

It's an unrepeatable, inspiring show which proves that great art doesn't take stacks of cash, just ideas and fearlessness. But, as Amanda Palmer will find, the money doesn't hurt either.

Tom Petty, a man who's parted his hair straight down the middle for 35 years, has always hung back a safe distance from the avant garde: far enough, of course, to slot seamlessly in among the ancients of The Traveling Wilburys (whose "Handle with Care" gets an outing tonight). It's hard, therefore, to believe it now, but for a long while – as far as 1979's Damn the Torpedoes, in fact – Tom Petty and the Heartbreakers were broadly considered to be part of punk.

After a solo career of sorts, Petty is once again backed by most of the surviving members of the Heartbreakers. More exciting for the layman, however, is the sudden appearance of Steve Winwood, who delivers two of his classics: The Spencer Davis Group's "Gimme Some Lovin'" and Blind Faith's "Can't Find My Way Home".

It's the more familiar hits, though, that get the big responses: "I Won't Back Down", "Don't Come Around Here No More", and above all "Free Fallin'", which gets a mass singalong on the verses, not just the chorus.

They encore with "American Girl", best known to Brits for its use in Silence of the Lambs. Republican Michele Bachmann tried to use it to launch her presidential bid, until the singer demanded that she stop. Tom Petty is no one's idea of a radical, but he ain't no conservative either.

Critic's Choice

The unbranded, unsponsored Hop Farm Festival brings headliners Peter Gabriel, Bob Dylan and Suede, and a supporting cast that includes British Sea Power, Dr John and Bruce Forsyth – yes, really – to Paddock Wood, Kent (Fri-Sun). Meanwhile, Ronnie Wood and Friends play Hammersmith Apollo, London (Sat). And Ronnie Wood has some very impressive friends.