As I was recently struck by Bill Callahan's cover of Kath Bloom's "My Baby Cries/The Breeze", I looked up a few covers that I find lovely. You'll find the originals by hovering your mouse-pointer over the titles of the originals, just under each video.

Radiohead cover The Smiths' "The Headmaster Ritual"

Jarvis Cocker covers David Bowie's "Space Oddity"

David Bowie and Moby cover Pixies' "Cactus"

Diamanda Galà¡s covers Blind Lemon Jefferson's "See That My Grave Is Kept Clean"

Ash covering ABBA's "Does Your Mother Know?"