Unless I've missed it no-one's mentioned This Mortal Coil - Song to the Siren (Tim Buckley) -
way better than the original.
I like the previously mentioned Blancmange/Human League/Schneider TM covers.
Also:
Danielle Dax - Tomorrow Never Knows (Beatles)
Cabaret Voltaire - Here She Comes Now (Lou Reed) & No Escape (Saxon)
Kode9 & The Spaceape - Sine, originally Sign of the Times - (Prince)
Age of Chance - Kiss (Prince)
Thanatos - That's The Way (I Like It) (KC & the Sunshine Band) (30s sample
here ) - possibly the most miserable cover version ever!