I would never claim that this is better than the original, but for me at least, Primal Scream's version of the 13th floor elevators song Slip Inside This House is the perfect cover because it retains the psychedelic 60's summer of love spirit of the original song while making it utterly contemporary (for the time it was released anyway, although to me it still sounds pretty fresh).
Did someone say bluegrass Radiohead covers? Chris Thile/The Punch Brothers have you covered. Here's a few selections. They have more if you dive into youtube.
Now look, I love the original. Freewheelin' is a great album and Don't Think Twice is a fantastic song. But 9 times out of 10, I'm choosing the Ness version because musically it matches the mood of the lyrics so much better.
Here is a cover of Taylor Swift's Love Story by black midi. Their bassist who doesn't often perform vocals sings here. And they added a saxophone solo!
While this may be a bit out of left field for most, I post this to commemorate the reaction that Goober and the Boys got from my friends and I the first time we heard it. We were loving the show already, but we all just burst out laughing, hooting and singing along when the boys started “Whole Lotta Love” by Led Zeppelin. Dan and the boys always put on an awesome show.
The Casual Dots are a trio formed in Washington, DC in 2002, featuring Christina Billotte (Slant 6, Quixotic, Autoclave) on vocals and guitar, Kathi Wilcox (Bikini Kill, Frumpies) on guitar and backing vocals, and Steve Dore (Snoozers, Deep Lust) on drums.
It's hard to beat Big Black's version "The Model" in this regard, but award for the most esoteric would have to be Atom and His Package covering The Mountain Goats "Going to Georgia", both great and totally different songs.