I'm not sure exactly what a breakout character is. Wikipedia has a list of breakout characters, but doesn't seem to define what it is. TV Tropes is all over the place.

I think the definition should have at least two aspects to it: first, it should be something of a surprise character, who wasn't set up to be the star; and: second, the character may or may not advance to the lead of the show, but should become very well-known in the popular culture of the time.

I think that these are among the biggest breakout characters in American comedy shows since I've been following TV:

Maynard G. Krebs (Doby Gillis)
Jose Jimenez (Make Room for Daddy)
Arnold Ziffell (Green Acres)
Pat Paulsen (Smothers Brothers Comedy Show)*
Tiny Tim (Rowan and Martin's Laugh-In)
J.J. Evans (Good Times)
Arthur Fonzarelli (Happy Days)
Miss Piggy (The Muppet Show)
Jim Ignatowski (Taxi)
Arnold Jackson (Different Strokes)
Alex P. Keaton (Family Ties)
Sophia Petrillo (The Golden Girls)
Steve Urkel (Family Matters)

*Pat Paulsen was sort of like Jerry Seinfeld - a real person playing a fictional character of the same name.

I have no idea if Bart or Homer Simpson is a breakout character of the Simpsons. My suspicion is that Bart was one in the beginning of the series, and then Homer became one later on.

I think a breakout character is one is only meant to be a background character or only meant to be on the show for a short time becomes so popular that they become regular characters and start to have storylines written for them. For example Jamie Farr was originally signed to play Cpl. Max Klinger in one episode of M*A*S*H but the character was so popular that he became a regular for all 11 seasons.

Two perfect examples would be Eddie Haskell from "Leave It To Beaver" and Gomer Pyle from "The Andy Griffith Show".

Steve Urkel is the perfect example of a breakout character. He was only supposed to be in one episode but everyone loved him that the show became all about him.

I know this isn't a sitcom, but the character of Ben Linus was only supposed to be on "Lost" for a few episodes, then they made him a regular that lasted to the end of the show.