The Dickies also did an excellent cover of Eep Op Ork, as well as the Gigantor theme.
I don’t want to be overly pedantic [sub](well, actually I do)[/sub] but we’re mixing things up a little here. Some of these songs were written expressly for the cartoons, like Sugar Sugar, while others were songs that pre-date the cartoons in which they were included. “Is You Is” was written by Louis Jordan and Bill Austin in 1944. “Why Don’t You Do Right” was written by Joe McCoy (as Ethilrist notes) in 1941. “Hello My Baby” was written in 1899, and “Please Don’t Talk About Me When I’m Gone” was written in 1930 by Bee Palmer, Sam Stept, and Sidney Clare. “The Great McCloskey Fight”, however, is a WB original. 
I was thinking more about songs written expressely for the cartoon but I wasn’t very clear. And everybody was having so much fun. . …
Some good ones mentioned so far…but it’s no contest: the Snorky episode (“There’s No Such Thing as a Sea Serpent”) of Beany & Cecil.
All of the songs in “One Froggy Evening” are brilliantly executed. I have had “The Michigan Rag” in my head for approximately 3.47% of my life. (Second only to Del Shannon’s “Runaway” at 4.13%)
As for the Simpson’s, l’ll nominate “See My Vest,” “Planet of the Apes–The Musical” and the song about the brothel whose name escapes me.
That would be “We Put the Spring in Springfield”.
I would have to say it’s a tie between three songs from Anastasia:
“Once Upon a December”, “Journey to the Past” and “At the Beginning”.
Probably “Once Upon a December”.
Yet another vote for Yakko’s World ( I have it on 2 cd’s)
“United States, Canada, Mexico, Panama, Haiti, Jamaica, Peru.
Republic Dominican, Cuba, Carribean, Greenland, El Salvadore too”
And “Monorail” by the cast of the Simpsons.
“3 is a magic number,
yes it is, it’s a magic number.
Somewhere in the ancient mystic trinity
you get 3 as a magic number.”
I loves me some schoolhouse rock
Actually, I think that was from Toy Story ** II**.
Regardless, it gets my vote, too.
For the not written for, but performed by, the cartoon section - “Why Don’t You Do Right?”
For the written for a cartoon section - movies- “Kiss The Girl,” (ballad), “Gaston” (ensemble number), “Everybody Wants to be a Cat,” (great song in the middle of a questionable film) and “The Siamese Cat Song” (they’re on screen for what, 2 minutes? And they steal the show with that one).
And for television - “Conjunction Junction”
I have to live up to the name once and a while. Besides, if you hum/sing loud enough, everyone will join in. What a way to start the weekend! Of course, I personally prefer (to the same tune) :
I am Evil Homer!
I am Evil Homer!
I am Evil Homer!
I am Evil Homer!
So easy to remember, so hard to forget.
LOL. I have no idea, but I was amused by this portion of your post. 
In addition to some already mentioned, I’ve always been partial to
“I Just Can’t Wait to Be King” from The Lion King
“God Help the Outcast” and “Hellfire” from Hunchback of Notre Dame.
“Hakuna Matata”
For all its faults Rock and Rule has some excellent songs;
songs by Cheap Trick /
Debbie Harry /
Lou Reed /
with Iggy Pop /
and a special performance by Earth, Wind and Fire
Send Love Through, Pain and Suffering…
The best song sung by a cartoon character?
That would be Trogdor - The Burninator.
This far, and no one’s mentioned the DOOM SONG sung by GIR in the first episode of Invader Zim? Oh the shame!
Another vote for “Kyle’s Mom’s a Bitch,” from South Park.
There’s a Yosemite Sam song that I sing to my dog (much to my husband’s chagrin):
Oh, I can't get along little doggie,
Oh, I can't even get one that's small,
Oh, I can't get along little doggie,
Oh, I can't get a doggie at all.
I love the music in all of the Looney Toons cartoons, especially Bugs.They fight, and fight,
And fight and fight and fight,
Fight, fight, fight,
Fight, fight, fight,
The Itchy and Scratchy Shoooowwwww!
Best Cartoon Song EVER!