Tim Minchin - Probably my favorite artist. Combines clever writing with very good music and awesome piano play. Inspires me to learn piano.
Ten foot cock and a few hundred virgins
Rock and roll Nerd
Flight of the Concords - two guitar players from New Zealand
generally more of a standup act, but still good.
The humans are dead
Hiphopopotamus vs. Rhymenoceros
Bo Burnham - the first music comic I started listening to. Has a rapping kind of style with lots of puns. Probably the most popular in terms of people that know him.
I'm Bo Yo
New Math
Bill Bailey - does more standup, and bar jokes then music writing but still has some very funny and talented stuff.
Love Ballad
Chaucer Pubbe Gagge - ok not a song, but its both hilarious and genious
Feel free to share your favorite music comedians.