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).
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.
While he's not technically a villain, the Janitor's verse in the middle of Dr. Cox's Rant Song certainly counts.
One Saturday Night Live "TV Funhouse" sketch had a fake preview for "Titey", a animated Titanic film produced by Disney. The villainous iceberg (voiced by Whoopi Goldberg) apparently gets her own villain song.
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.
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.
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 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.
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.