dresden dolls - pretty in pink (psychedelic furs)
The Dresden Dolls do some good covers. I really liked the versions of "Amsterdam" (Jacques Brel), "War Pigs" (Black Sabbath), and "Such Great Heights" (The Postal Service) they did when I saw them live. However, I saw a video of them doing a cover of "Two-Headed Boy" (Neutral Milk Hotel) and it made me cringe. I'm not sure I've ever heard an acceptable cover of an NMH song. Wait, no, that's a lie... The Mountain Goats. But that's different. I am currently trying, and failing, to find this furs cover online. Dammit.
neutral milk hotel - i love how you love me (phil spector)
This cover is fucking beautiful. God yes.
stars - this charming man (the smiths)
I just downloaded this to give it a listen and it's quite lovely, actually. I like Stars, and that's one of my favorite Smiths songs (the fast version with the Morrissey Squeal, more so than the slower one) aaand somehow they go together well.
In addition:
David Bowie - See Emily Play (Pink Floyd)
Pavement - The Killing Moon (Echo and the Bunnymen) yes yes yes yes whoever mentioned this one.
Bauhaus - Ziggy Stardust (Bowie, dude) yes yes yes again to whoever mentioned THIS one
Nouvelle Vague - Human Fly (The Cramps)
The Pixies - Head On (The Jesus and Mary Chain)
All of Seu Jorge's Bowie covers are amazing.
Also, I heard a cover of Life on Mars? recently, by some (Norwegian? Swedish? Icelandic? Something...) chick whose name I cant remember, but it was quite good.