Random favorites of mine:
Doc Hopper - "867-5309 (Jenny)" by Tommy Tutone
Screaming Trees - "What Goes On" by VU
Neko Case - "Train From Kansas City" by (I think) The Shirelles
Tommy Keene - "Einstein's Day" by Mission Of Burma
Big Dipper - "The Girl Who Lives On Heaven Hill" by nem Husker Dudes
The Reigning Sound - "Brown Paper Sack" (composer was Larry Rogers, and I've heard an old version, but I'm not sure if it was Rogers performing the tune under his name or not; "last edit" coming up...)
Soul Asylum (screw you in advance, detractors) - "Barstool Blues" by Neil Young
The Queers - "It's Cold Outside" by The Choir
Lyres - "Seven" by The Sevens
Lyres - "No Reason To Complain" by The Alarm Clocks
(just a couple of examples of the great obscure covers that form the backbone of this band's recorded output)
The Rolling Stones - "That's How Strong My Love Is" by Otis Redding
Steel Pole Bathtub - "Surrender" by Cheap Trick
The Dogmatics - "King Size Cigarette" by The Law (a pre-Scruffy The Cat Charlie Chesterman outfit)