The Incompetent Mayor Pain (The Quimby): Incompetence over wickedness. The mayor is openly incompetent and his/her actions are clearly due to a lack of judgment. On a darker note, this incompetence could cause (or have already caused) various disasters.
An Orange Islands Pokémon episode had one of these. When a mystery creature is wandering the sewers on his island, he intends to kill it rather than capture it, as he's worried its existence will affect his status as mayor. Turns out it was his Bulbasaur he abandoned as a child. Poor thing was still happy to see him.
Hamilton Hill from the Batman comics was an Evil Mayor Pain; a competent politician, but completely in the pocket of crime boss Rupert Thorne.
He was succeeded in office by Armand Krol, who was an Incompetent Mayor Pain.
In Batman Earth One, Mayor Oswald Cobblepot (The Penguin in the regular timeline) is Mayor of Gotham. He's also a crime boss who uses a Serial Killer named The Birthday Boy as his own personal hitman, supplying him with teenaged girls as payment.
The Mayor in Cloudy with a Chance of Meatballs starts off as just a jerk, but once the food starts falling, he will do anything to keep it coming, since his town is now getting notoriety and he is quite a glutton. He ends up pushing Flint to keep making the food, even when told it's not safe. And when the inevitable happens, he quickly shifts the blame to Flint.
Mayor Phlegmming in Osmosis Jones, is as incompetent as they come. Its his idiotic decisions that are the reason why Frank is such a slob and he cares more about his re-election than the lethal threat that would and almost did kill Frank.
As there's no evidence he has any personal planning or administrative skills, and Grice's personal and family histories must be consistent, he probably really is an incompetent governor. They show up a lot in the setting, and tend to stay in place until they impede somebody important who can have them killed.
Once Upon a Time: Regina Mills, aka The Evil Queen, is definitely the evil version.
Criminal Minds: Clark Preston from "A Thin Line" was going to become one, had the BAU not found out about the murders he had planned.
"The Mayor" by Rasputina revolves around a mayor described as a lunatic paranoid.
Leonard Zachary of the Shanafelt City Council from TRU-Life Adventures is this in all but title. Getting the city to restore the position of mayor is high on his list of priorities.
In Daryl and Susie, Daryl the dragon has a town in his head, run by an Evil Mayor Pain (based loosely on the Mayor from Buffy). This Mayor incessantly tries to get Daryl upset, for doing so makes monsters appear from Daryl's head, and collaborates with Daryl's arch-nemesis Aosoth before becoming The Starscream.
Decembersville has Chimabell, who is most definitely the evil variant.
Mayor Pamela Winchell from Welcome To Night Vale. She's a strange mix of both the "evil" variant and "incompetent" variant, although she's not stupid and it's difficult to peg her as out and out evil. Given the way she's described, it seems more like she's just totally insane.
Mayor Joe Quimby from The Simpsons provides the name for the Incompetent Mayor Pain. An outright corrupt, opportunistic, embarrassing, sleazy politician who couldn't care less for Springfield.
Sideshow Bob may count as the evil variant during his brief time as mayor.
Vlad Masters/Vlad Plasmius eventually becomes this in Danny Phantom, abusing his authority to antagonize Danny.
Mayor from Powerpuff Girls is so incompetent, he somehow managed to kill Navi when playing Ocarina of Time.
After the events of the first season finale of Scooby-Doo! Mystery Incorporated, Mayor Jones is a definite Evil Mayor Pain. In that one episode we learn that he moved to Crystal Cove only to look for treasure, blackmailed the original Mystery Inc. into leaving, double-crossed Professor Pericles, dressed up as the Freak of Crystal Cove, and to top it off, concealed the true identity of Fred's real parents while raising him as his own son.
From The Oblongs, the town of Hill Valley is governed by Mayor Johnny "The Mayor" Bledsoe, a masked wrestler (talks like an American pro wrestler, wears a luchador mask with a nice suit). He's a good balance of evil and incompetence, like all authority figures in the series.