The Spanish sitcom "Aída" has an example in it's episode number 100. In this episode the character Mauricio Colmenero, a fascist and greedy innkeeper, sings "Soy un cabrón" in this song he boasts that he has no moral and that he will do anything for money. This song is played after he explains to his employees that he made a pact with the mayor in which he will gain a lot of money if he betrays his neighbors cheating them to sing a contract where they agree to leave their homes and let their district to be demolished to build a highway.
Arrested Development invokes this in its fourth season, when Lucille (quite possibly the most evil member of the family) is forced to write one from the heart to audition for a Fantastic Four play. The results are somewhat terrifying.
In the category of "not sung by the villain" is Black Adder's title song, which has many variations throughout the life of the series and associated specials. The lyrics both praise and lambaste the titular Villain Protagonist in the first and the second season.
And of course, "Voodoo Child" from the episode before. "Here come the drums, here come the drums..."
Emmet Otter's Jug-Band Christmas has the Riverbottom Nightmare Band singing an eponymous song. Lyrics include "We don't brush our teeth, 'cuz a toothache can help us stay mean" and "We break up your place / We are dangerous when we are near". The song actually fits into the plot because it is an entry in the local Talent Contest.
In the Musical Episode of Even Stevens, we have "Master of the Gym", sung by Coach Tugnut as he forces the boys to participate in a sadistic obstacle course. In true Villain Song tradition, it's one of the best songs in the episode. Although it's also a subversion, since it ends with the boys standing up against him and forcing him to participate in it instead.
Ironically, Richard III- infamously immortalized by Shakespeare as the arch-villain Richard III- gets a sweet, longing ballad in which he proves himself to have been quite a nice guy.
"Bloody" Mary Tudor also gets a musical chance to explain that she really was trying to be good (yes, back then that could easily include burning "heretics" at the stake) and wasn't so much unsuccessful as pathetically naive and unfortunate.
"VikEngland" is about the many developments the Vikings brought to Britain.
The Lallakki song Kill the Humans in BBC series Hyperdrive.
With the recent news that Stefan Karl, the actor who played Robbie Rotten, is currently undergoing chemo for pancreatic cancer, a GoFundMe page has been setup for him and some Youtubers have used their videos as a donation platform.
MacGyver even has one of these in the Season 5 premiere, 'Tough Boys', appropriately called the Tough Boys Rap. Of course, it only works if you believe the Tough Boys are the episode's Designated Villains, because they're just vigilantes trying to end their neighborhood's crack problem. Nevertheless, the song sounds pretty cool.
"Love Doesn't Stand A Chance", the Evil Queen's number about how nothing's going to stop her from crushing the Charmings.
"Revenge Will Be Mine", Hook's song about his intent to hunt down and kill Rumplestilskin.
"Wicked Always Wins", Zelena's melody about how she's going to watch Regina fail and then come out on top of her.
Subverted by Rumple, who Trolls Regina into thinking he's about to sing (with a melodic version of his leitmotif even appearing for a moment), then reveals that the spell didn't work on him because "The Dark One doesn't sing". Ironic and something of a shame, since Robert Carlyle actually can sing quite well.
Scrubs While he's not technically a villain, the Janitor's verse in the middle of Dr. Cox's Rant Song certainly counts.
Super Sentai has a few examples: Himitsu Sentai Goranger has Akuma no Kurojuujigun, JAKQ Dengekitai has "Iron Claw's theme", Kousoku Sentai Turboranger has Oola no Urami, Kyoryu Sentai Zyuranger has Bandora's theme, "Dolla! Theme of Witch Bandora". And compare the opening theme to Gekisou Sentai Carranger (seen here) with the theme song to The Psycho Rangers of that series, Bousou Sentai Zokuranger (here).