Sideshow Bob sings a lot, but he had a good Villain Song in "The Great Louse Detective" called "The Very Reason that I Live", where he explained to Bart that he spared him because killing Bart would put him in an unbearable Victory Is Boring situation.
The SpongeBob SquarePants episode "F.U.N.", in which Plankton sings a demented and deliciously villainous reimagining of Spongebob's friendship song "F.U.N." - "F is for fire that burns down the whole town, U is for uranium - bombs!, N is for no survivors WHEN YOU—..."
"Plankton! Those things aren't what fun is about!"
Plankton gets a full-blown Villain Song in "Atlantis Squarepantis", simply called "Plankton's Song".
Notable in that he is the ONLY villain in the entire show to get one.
Slade of all people is rather put off by this; see below.
He gets another one in the DC Nation Teen Titans short "Mayhem at First Sight", where he sings about his love for Mother Mae-Eye and the carnage they'll wreak together.
Grandma Stuffum's rock opera, "Slamwich", in the Codename: Kids Next Door episode F.O.O.D.F.I.T.E. GWAR did the music for the episode, and the number is a combined variation of their songs "The Private Pain of Techno Destructo" and "Gor-Gor".
Also, Stickybeard's "Licorice, Licorice" from L.I.C.O.R.I.C.E.
Discord, despite no longer being a "villain" as of season 4 (although still a Trickster and a minor antagonist), finally gets his own song in "Three's a Crowd," and it's every bit as over the top and hammy as you'd expect a villain song from him to be.
In her first appearance, her song "Take The Easy Way Out" is used to tempt Charlie AND insult her cousin Annabelle.
The Backyardigans: the first superhero-themed episode, Race to the Tower of Power, had "We Are Supervillains" and "Key to the World". (The performers (Pablo and Tyrone) are villains just for that episode, but it still counts.)
In the episode Cops and Robots, there's We Are Bad Bots sung by Pablo (as the Pablonator) and Tasha (as T-800).
From the same episode as "One of Us" comes "The Mirror Never Lies", a preceding song by Kaptain Skurvy where he convinces Donkey Kong that he's a Kremling Pirate. Klaptrap, although he's more of a neutral than anything, gets a song of his own called Someone Get My Teeth Back!.
Of particular mention should be "Become a Bad Guy", In which we get some amusing anecdotes of just a few of the villainous things King K. Rool does whilst Candy helps Donkey Kong try to figure out how he can play the part of a villain.
The Rankin-Bass stop-motion Christmas special The Year Without a Santa Claus has two ... well, villains may not be quite the right word, precisely; they're more like obstacles who are mostly just narcissistic instead of being outright evil ... who each get a variant of what's basically the same song, "Heat Miser/Snow Miser". It's by far the most catchy tune in the show.
The sequel, A Miser Brothers' Christmas, feature the song "My Kind of Christmas" sung by North Wind, who plans on ridding the world of Santa Claus so he can take his place for fame.
From the 1970 animated adaptation of Horton Hears a Who!: "We're the Wickersham Brothers, we're onto your plot — pretending you're talking to Whos who are not! It's a deep, dire, evil, political plot — pretending you're talking to Whos, who are not!"
My name is Crabby Appleton And I am really awful, It titillates me funny bone To do a deed unlawful. I'm fond of gloom, impending doom, I think good deeds are sappy, I laugh with glee, it pleases me When everyone's unhappy.
Magnus Hammersmith and the Metal Masked Assassin in Metalocalypse The Doom Star Requiem have "Magnus and the Assassin", with Magnus torturing Toki as he and the Assassin talk of their plots to exact vengeance upon Dethklok. The song focuses mainly on Magnus, though.
Bump in the Night has featured a villain song for both Destructo and the alien duo Sleemoth and Gloog. Destructo sings "I'm Bigger than You" in one of the Karaoke Cafe segments, where he brags about how everyone should do what he says because he is tall and powerful. Sleemoth and Gloog sing a song called "Invaders" in the episode "It Sang from Beyond the Stars", where they mention their plans for world conquest while pretending to be a rock band.