Another example from JAM Project: The Gate of the Hell, the opening song for the Mazinkaiser movie. The song starts to appear just after Lori and Loru gives their lives to ensure Koji's survival. After a mourning and a This Is Unforgivable kind-of phrase from Koji's part, cue opening sequence, which displays how the other mechas are unable to deal with the new threat, while Koji, Sayaka and the Professor Morimori are trying to return to Japan for the eponymous mecha.
Plenty in Mermaid Melody Pichi Pichi Pitch. The first season only had two. "Kuro no Kyousokyoku ~concerto~" was a dark rock revelry filled with Minion Shipping and singing about breaking the bonds between the mermaids; appropriately enough, it's introduced around the same time as Caren, who threatens to tear the group apart, and it's eventually defeated by a song called "Kizuna" (Bonds). "Return to the Sea" hinted at Sara's motivations and got a Triumphant Reprise near the end. The second season gave the Black Beauty Sisters another song, "Yami no Baroque", much like their previous song, and gave each of the new villains their own songs where they sing about their powers (Lady Bat about hypnotism and vampirism, Lanhua about her instant minions, Alala about her Lotus-Eater Machine stars) or their loneliness (Michel's "Tsubasa wo Daite", which also hinted at his connection with Michal because her "Ashita wa Mienakute" used some of the same music).
Being a Musical Assassin, the Gag Dub gave Etemon a pair of songs in the English translation as well: "Yeah! (The Coolest Cat There Is)" in Episode 17 and "Big-Time Digi-Destined Blues" when he comes Back from the Dead in Episode 46.
Team Rocket from Pokémon had their own self-sung theme, at least on one of the American soundtrack albums: "Double Trouble". Though, like some examples above, this also falls into the Quirky Miniboss Squad territory (though "The Boss" does chip in a few lines, too).
They get their own song in the Japanese version, too. Two of them, in fact: the bombastic "Rokketo-Dan yo Eien Ni" (Team Rocket Forever) and the more comical and fast-paced "Maemuki, Roketto-dan!" (Look Ahead, Team Rocket!)
They also get a song in the show itself when they decide one day that they're tired of doing their motto over and over again. Ironically, the song is only ever heard again once, and then only in part.
One of the Japanese soundtracks had the song "Ware wa Collector / I am a collector" for Lawrence III AKA the collector from Pokémon 2000.
And in the filler Doma Arc (a.k.a. Waking the Dragons), the main tool of the villains is a Field Spell card called the Seal of Orichalcos, which makes it so the loser of the Duel loses their soul. In the 4Kids Entertainment dub, the same music plays every time it's activated; surprisingly enough, it's easily one of the most epic themes in the series, as can be heard here.
Sailor Moon doesn't usually give the villains image songs, but they did for Galaxia. Her song, Golden Queen Galaxia is equal parts bitchin' guitar solo, jazz-tempo piano and evil harpsichord. Pretty badass overall.
Queen Beryl also got a song as well (albeit it wasn't sung by her voice actress) but it is nevertheless meant to be Beryl's song about stealing energy.
The second CD then followed this up by giving ones for #18 and Cell as well.
The Japanese band, Maximum the Hormone, wrote this song about Freeza.
Rip Van Winkle from the fourth Hellsing OVA was fond of singing the German opera Der Freischütz. The YouTube clips seem to have her singing the hero's part — O diese Sonne!. And when Alucard attacks her in the manga, he sings the demon's part.
As Misa Amane (like her crush/master Kira) is a Knight Templar extremist with genuine "good guys" as enemies, her song in the Death Note episode "Silence" is technically a Villain Song of sorts - it focuses on hers and fellow villain Kira's support for each other (Kira is even mentioned as "God" in the song). Since Misa is no Card-Carrying Villain, the song is a soft, sentimental, simple ballad, and sounds like the thing that would normally be sung by a non-evil character (both Misa and Light genuinely believe that they are doing the right thing for the world). Not to mention the fact that she's singing it while killing people.
Since "Mr. Grinch" counts, one may submit that the entire first opening theme to Death Note is, in fact, a villain song. When the lyrics are translated, they are essentially Light's philosophy and goal touted out for the world to see as the show begins. The difference, though, is that "Mr. Grinch" is sung by the Grinch himself.
What do you want? I'll snatch it from you What is precious to you? I'll shatter it completely Where are you going? I'll block your way
From the same series, Yui's image song, "Aoi Arashi", wherein she sings of her desire to take revenge on Miaka and Tamahome:
I'll make it so / you alone will never be happy. * The Black Four from Kimba the White Lion have a catchy and somewhat surreal number about "doin' dirty business in style".
Big Bad Miyo Takano of Higurashi no Naku Koro ni has a rather fitting Villain song in the Higurashi no Naku Koro ni Kai ~Character Case Book~ Vol. 2, Bon~Karma, with evil laughing, death threats and all.
The Mobile Suit Gundam main rival, Char Aznable is given "Char ga Kuru" as his theme music, though it is not in the anime.
Actually, the song is used in the anime. In episode 13, "Coming Home", the Federation soldiers occupying Amuro's mom's house are listening to the song on the radio. In the 2000 Japanese redub of the first movie, the scene where the White Base crew are vacationing on a beach while Amuro is off visiting his mom also has them listening to the same song on the radio.
Gundam ZZ has Silent Voice, sung from Haman's viewpoint towards Judau and company, used as the 2nd opening.
In Rock Muscial Bleach, Aizen, Ichimaru and Tosen have one, spoken in Japanese with a mix of Gratuitous English, "Catharsis of Eternity."
There's also the Bleach Beat Collections, three of which (at least) have the voice actor for the character singing two songs pertaining to their character. "Kyouka Suigetsu" by Aizen's is particularly haunting, while Ulquiorra's song is apt to his personality by title; "Crush the World Dow N". Grimmjow also has two songs, one titled "Br Ea K" and the other as "Six Feelings".
While there are no villains singing in Kinnikuman itself, EVERY villain has a villain song released on soundtrack. The villains themselves don't sing them admittedly, but it IS written as if they were the ones singing them, and they do give intros to the song, and sometimes even have spoken lines in them.
Odd example within the fact that, heard singularly, neither song is particularly villainous: Meikyuu Butterfly tells of both her love for Ikuto (And also has some lines referring towards the fact that its incest.), and, upon a related note, how the easter company has trapped her by making threats to harm Ikuto to force her to cooperate (Them both leaving results within the same being done to their mother.), whereas Black Diamond is something that only relates to the topic with the above mentioned question, otherwise not relating towards anything involving Utau her - self at all.
Nerima Daikon Brothers, being a musical anime, a villain song was bound to pop up. You have two songs from the corrupt fortune teller set to the same tune and a lot of reappearing tunes are fit onto the villain of the week, from abusive host boys to a corrupt nurse. In the final few episodes the Prime Minister gets his own about his political ideals playing in his floating fortress, but it wasn't translated in English because of how the audio was recorded. They lampshade this in the dub.
The song "Voodoo Kingdom" performed by the rap group SOUL'd OUT for the Phantom Blood movie is this to Dio Brando.