A reader writes:
To me this cover of "Wonderwall" by the The Mike Flowers Pops is the classic example. The video actually got significant airplay on MuchMusic up here in Canada when it came out, which was around the time of the original.
I can hold my silence no longer. Of all the genre-shifting covers, I have never heard anything more impressive than Bruce Lash's bossa nova-style cover of the penultimately poppy "Psycho Killer" by Talking Heads.
Not the creation of a random language generator, I promise: Apocalyptica Four Finnish cellists who have a whole album of Metallica covers.