Ooh, ooh - my favourite subject!
Somebody beat me to Tatu's unironic version of How Soon Is Now?
The Cardigans, in the early twee phase of their career did Iron Man, Sabbath Bloody Sabbath and a sun-drenched and jazzy "The Boys Are Back In Town"
Galaxie 500 did a storming version of the Pistols' "Submission" (good covers band - see "Here She Comes Now", "Listen The Snow Is Falling" and "Isn't It A Pity?")
Similarly, Low are good at this kind of thing - as well as Transmission, they did "Last Night I Dreamt That Somebody Loved Me" and an entire set of Misfits songs as an encore (hidden tracks on the box set).
Calexico's mariachi-tinged version of the Minituemen's "Corona"
The Manic Street Preachers's "Raindrops Keep Falling On My Head"
Gillian Welch and David Rawlings doing Radiohead's Black Star - especially live
Big Black doing Kraftwerk's The Model
Ryan Adams's version of Always On My Mind - or even Wonderwall!
I even like The Cowboy Junkies' version of Sweet Jane, even though they miss out a crucial chord.