This came up the other day in another thread, but The Clash's cover of I Fought The Law is epic. Ugly Kid Joe did a really good job on Cat's In The Cradle, but that may be more of a personal preference than widely loved cover.

I find a lot of people actually have no idea when a song is covered. I mean, sure if the original was played by The Beatles then most will know, but if it was done by someone less famous decades later, then most won't.

You could still do a trance cover of something. :P It doesn't have to be improvised, that was my point.

On a side note, my taste totally the opposite of yours (I dislike computer generated music and don't even like it when a studio recording sounds too 'arranged'). I went to see a folky hero of mine, Robin Williamson, and at one point in the gig he did a small medley of psychedelic-y Beatles songs (Tomorrow Never Knows, The Inner Light and I think there might have been another one I'm forgetting). It was amazing!

If music isn't live, then it's dead (ho ho ho).

Laibach's cover of Opus - Live is Life is filled with singular badassity: Laibach - Opus Dei

Better than the Original Shining Star

His entire album titled "The Man Comes Around" consists of covers, most of them brilliant. My own favourite is Personal Jesus.

I think the best covers are the ones that do something entirely different with the song than the original version did. Here's a couple of covers that take the songs they cover in a completely new direction.

Apocalyptica - Nothing Else Matters (Metallica)

Me First And The Gimme Gimmes - Hello (Lionel Ritchie)

If you're into punk covers, Me First And The Gimme Gimmes deliver every time, by the way. I highly recommend them.

Finally, this one isn't exactly all that different from the original, except in the way it's performed, but it's absolutely fantastic: AudioFeels feat. K8 - Crazy (Gnarls Barkley) The ultimate cover of In My Life has to be Sean Connery's (please ignore the silly video, and just listen).

Frankenstein - Overkill (Edgar Winter Group)
In-A-Gadda-Da-Vida - Slayer (Iron Butterfly)
Raining Blood - Vader (Slayer)
War Pigs - Sacred Reich (Black Sabbath)
You Really Got Me - Van Halen (Ray Davies)
Bitch - Exodus (The Rolling Stones)
Dirty Deeds Done Dirt Cheap - Exodus (AC/DC)
Low Rider - Exodus (War)
Smoke on the Water - Metallium (Deep Purple)
Am I Evil - Metallica (Diamond Head)
These Boots - Megadeth (Nancy Sinatra) Not a word for word cover but whatever
No More Mr. Nice Guy - Megadeth (Alice Cooper)
Paranoid - Megadeth (Black Sabbath)

I take your Cake, and raise you a Victoria Faiella.

Also, Celldweller - Tragedy (Bee Gees).