Remember the song Anything Anything by Dramarama? If you live in LA and listen to KROQ, eventually you will hear this song. As I do sometimes, I found a version on Soundcloud and investigated covers on YouTube. Here's what I found:

0. The original. Vocals are believable.
1. Vale Cover.  Instrumental part is pretty cool, vocals not so much.
2. Lucky Boys Confusion. Compelling, nice work
3. Beach Slang. Punky, brings something new but pretty close to original

I think groups want to avoid making too much fun of the original in a cover, like in the Lemonheads' cover of Mrs. Robinson, which they say Paul Simon hated.  Sorry Paul, once you discover and release a song into the wild, it's no longer entirely yours.  Perhaps this is why Prince had so much unreleased material locked away.

"See also" Maps by Yeah Yeah Yeahs, How Soon is Now by Smiths, and Joey by Concrete Blonde (probably my favorite of the bunch, no covers of which do justice to the original vocals).

If I get into a mood looking for covers again I'll post more here.  I'm continually amazed by the quality out there.  Hope not too much of it disappears over time.