You could throw a dart at a Mrs. Miller album and come up with a good answer to this thread but I'll just go with her doing Petula Clark's Downtown.
From another great album - 20 Extraordinary Renditions, which is 20 different cover versions of "The Idiot Bastard Son" Here's a cool a cappella one by a group called Die Beistelltische
This is a similar album, 21 Burnt Weeny Sandwiches, though it's not all the same tune, and it's a lot more loosely interpretive. Todd Grubbs Group - "Themes from Burnt Weeny Sandwich"
Shonen Knife—I’m on Top of the World (Carpenters). Nico—These Days. Vassar Clements—White Room (killer bluegrass version).