In the Bleach fandom, Humble Shopkeeper has one of the characters declaring she wants a villain song to "underline [her] bad-ass-ery". It involves the line "I'll paint the moon red with his blood", a chorus of wolves and evil laughter. Since it takes place in a realistic setting, it's pretty much as weird as it sounds.
Cyros from The Legend Of Spyro: A New Dawn gets "Into The Great Unknown", which she uses to scare her brother Pyrus into aiding her Evil Plan. Later on, she gets "Let Your Secrets Out", where she uses her powers to cause Cynder and Ember's hidden feelings to come to the surface. She also gets a short spell later on that transfers Empress Tyrania's energy into a jewel she can use. Justified in that she is an egotistical music lover and a Evil Diva who uses songs to cast spells.
Cyros gets one more during the Finale called "Welcome To Your Final Show", which is pretty much a straight up Villain Song about how she intends to kill her brother and Ember, all the while using her Musical Assassin powers to try and do so. She was Drunk on the Dark Side at the time, so naturally she's more hammy than usual.
Decks Fall Everyone Dies plays this straight and sort of subverts it. Duke gets a villain song in which he gloats about how he reigns over everyone using dice games and sexiness, but in the very same song, he hints at a nagging sense of guilt over it.
In the Pony POV Series, Princess Gaia, Fluttershy's Superpowered Evil Side and the second Big Bad gets two. The first is a ponyified version of "Come Little Children" from Hocus Pocus and the second is a Dark Reprise of Fluttershy's "So Many Wonders" song, telling how her world is safe and perfect and no one will ever have to leave it... And will forget their old lives entirely. It also serves a purpose, as she's an Evil Diva who can trap ponies through song.
Discord has gotten several. His father, Havoc got one called "Here's To Chaos" as well.
The same author also wrote one for Loneliness, Trixie's Enemy Within. See it here.
This piece of Recursive Fanfiction gives one to Nightmare Granfalloon, Pinkie's potential Superpowered Evil Side. Fittingly, the song is basically a psychotic version of Pinkie's "Smile, Smile, Smile". While they didn't write the fanfic, the same person who wrote the above two also wrote this one.
Said author has also recently written one for Strife, Discord's sister and the Final Boss of the "Generational Transitions" arc.
Earlier in "Off Da Rails" Dark World!Discord and Fluttercruel, in Sparkler's body, have one inspired by Joker's Song called "Disharmonious Duet" about the love of sadistic games both share though given The Reveal, Discord was likely only singing along for her sake.
Cruel Eyes/D___t's Shadow of Existence gets one in Part 6 of Bright Eyes' Gaiden as well in the form of a Dark Reprise of Bright Eyes' "In the Forest" song from My Little Pony Tales in an attempt to convince Bright Eyes she subconsciously wished for the apocalypse to happen, in an attempt to drive her over the Despair Event Horizon so he could consume her infant Concept and become real again.
Chrysalis, being a diva (but not an Evil Diva), gets multiple songs throughout her Origins Episode and the Wedding Arc, including at least one variant of "This Day Aria" from canon.
In one episode of the noncanon "Pinkie Pie's Fourth Wall Breaking Verity Show" (what amounts to a series of Just for Fun chapters composed of audience contributions to tide over while chapters are being worked on), a revived Hydia gets a version of "Wicked Ways" from 'Lion of Oz' while fighting Twilight and Shining Armor with an army of monsters. It's also referenced that Hydia had the original Villain Song in MLP.
King Sombra does not get many fanmade Villain Songs because he is more like a force of nature than a proper character. However, Crossing The Line is very powerful and intimidating, fitting the evil king.
Eurobeat Brony's "Batty" is one of these for Flutterbat.
"Through My Eyes" gets a reprise in a later chapter, where Checker details her motives for trying to ruin Trixie. She's extremely self-centered, and has always been jealous whenever anypony (especially their father) paid more attention to Trixie then to her, so is determined to ruin her. In case you hadn't realized it yet, both versions of the song really sell the fact that she's a sociopath.
She gets another song in chapter ten called "Checkmate" where, following her Villainous Breakdown, she declares that she's not finished with Trixie yet and will ruin her. It gets a reprise at the end of the chapter after she's successfully forced a Sadistic Choice on Trixie that leaves her with nothing.
Her part of the group song "The World I Want To Live In" in chapter 11.
Inner Demons incorporates Adam Lambart's "For Your Entertainment" as a villain song for the Face Heel Turned Sweetie Belle, who sings it as she's using her Magic Music to brainwashevery stallion in Fillydelphia and sending them after the heroes.
An interesting example occurs in Chapter 2 of In Brightest Day. Right after gaining the Green Lantern Ring, Applejack is confronted by Ruby, Mitta, and a recently corrupted Granny Smith. During the fight the trio proceeds to sing "With Me" to further hammer in how hopeless her situation is and how she should just surrender. However at certain points of the song Applejack sings her own lyrics during the fight, and eventually takes over the entire song near the end, having it pertain to the fact that she won't surrender as she successfully defeats the trio.
The majority of "Carnival Cat vs Ponyville", a song centering around the title character's attack on Ponyville and her attempts to convince Big Macintosh to be her husband, for the Carnival Cat herself. The only parts not song by her are by the foals she's brought under her power and the Bearers facing off and defeating her with the Elements of Harmony.
The other is a parody of the same song, called "Will You Help Me Hide a Body?". A longer adaption of a one-verse version, this song involves Anna becoming a serial killer, likely because of the accident messing with her head, and eventually killing her parents, implying that she plans to kill Elsa next. If the comments on the video are anything to go by, another song parody and a fanfic set in that continuity are set to follow.