I fucking hate REM, but I think the covers of 'Nightswimming' and 'The Sidewinder Sleeps Tonite' recorded by The Wrens and Oxford Collapse respectively for the Stereogum tribute to Automatic For The People are both AWESOME! So they would be two of mine.
The cover which inspired me to start this thread is one I've just recently discovered though. It's a cover of 'You Are My Sunshine', and it's by a little band called THE FIRE SHOW. I'm going to go right ahead and say: best cover ever.
Other good covers:
Cobras Cobras Cobras - The Chain (Fleetwood Mac, obvs)
Pale Young Gentlemen - Paper Planes (MIA, obvs!)
Elliott Smith - These Days (Jackson Browne - he only ever did it live though)
Me - Steal My Sunshine (Len - it's bloody brilliant, I tell you)
On a related note, I'd also quite badly like to cover 'Young Shields' by Casiotone for the Painfully Alone and 'Revelator' by Gillian Welch, but I need a full band to do so as I've envisioned in my mind. And I'd really like to here David Frost covering 'Hey' by the Pixies. Or any song ever. I might come back and post some more later.