Cover songs that are better than the originals.

Diana Krall's "A Case of You".

Martin Carthy (one of Steeleye Span original members) did a acoustic cover of The BeeGees "New York Minining Disaster".  He makes it sound like an old English folk song. Brilliant.

Winger - Purple Haze
KEEL - Because the Night
Motley Crue - Smokin in the boys room
Twisted Sister - Leader of the Pack

Two by Ben Folds:
Common People (original by Pulp) on William Shatner's "Has Been" Album
Bitches Ain't Shit (original by Dr. Dre) from "supersunnyspeedgraphic, the lp"

NRBQ's cover of Johnny Cash's "Get Rythm" - so good that Johnny started to play it like them after hearing it

Kat Edmonson's version of Brian Wilson's "I Just Wasn't Made for These Times" is not necessarily better than the Beach Boys' original but it is great in its own way.

John Prine's cover of Blaze Foley's "Clay Pigeons" from "Fair & Square"

Glen Campbell's cover of Paul Westerberg's "Ghost On the Canvas" (Paul's recording is almost unlistenable - Glen's is genius).

There's been many mentions of "Sweet Jane". My vote is for Mott The Hoople's version.