Impressionists are a type of singing comedian who do celebrity impersonations. A true impressionist tries to vocally mimic the singer they are parodying. Jimmy Fallon does an excellent Neil Young and Bob Dylan, among others. Check out celebrity impressionist Matt Jernigan:
Song Parodists twist real song lyrics into humorous variations. You can be a parodist without being an impressionist. Popular song parodists include Tim Hawkins. You can also parody a whole genre of music, such as Jack Black does with Tenacious D. The most notable example of a song parodist is "Weird Al" Yankovic:
Funny Singer/Songwriters compose original melodies and lyrics. There are many famous examples such as Ray Stevens or Flight of the Conchords, but I'll use myself as an example because... well, it's my blog. Check out funny singer/songwriter Larry Weaver:
Song and Dance comedians may be a dying breed of performer. Legends like Dean Martin and Sammy Davis Jr. as "The Rat Pack" seamlessly blended traditional (serious) songs with comedy and dance. Check out this clip from way back in the Sixties:
There is another type of comedian who uses music as a backdrop to tell one-liners or stories. Contemporary examples include Demetri Martin and Zach Galifianakis, who both use the guitar and piano respectively as background music as they tell their jokes. I'm not sure what to call this group. Anyone have suggestions?
What other types of singing comedians can you think of?
Wikipedia has a good, but not exhaustive, list of musical comedians.
And I have a good listing of singing comedians for hire.