One of my favorite 1980's comediennes of all time is Sandra Bernhard. She now had a radio show called :"Sandyland," but back in the day (in the 1980's) she was well know for her own style of irreverent humor, on the "David Letterman Show." She also palled around a lot with Madonna. She got her first acting job playing in Martin Scorsese's "King of Comedy" in 1982. She was lauded for her comedic acting and also singing.
Here are two snippets taken from the show in 1984:
Here Sandra sings a bit as well: