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One step up from the Cowboy Cop who enters Hot Pursuit with no regard for their own safety or anyone else's, here you have the police who have the equipment, the powers, or whatever else that puts them above and beyond the capabilities of the average police officer. This is common to the point where one of the Stock Superhero Day Jobs is police work, though they don't always do both at once.
Often takes the form of a Superhero who works for the police, a Cyborg, or a Humongous Mecha. A particularly Badass Normal can qualify if the criminals they chase are superpowered and they are not.
Overlaps with the Vampire Detective.
Not to be confused with the Jackie Chan movie Super Cop (whose tagline provides the page quote).
Averted with Patlabor. They could be Supercops, but regulations have them stuck in the station until a crook brings out his own Humongous Mecha. For regular criminals they have to go on foot just like all other cops, and face the same obstacles.
Dominion Tank Police probably counts. I mean, these are cops... with TANKS. Really BIG tanks, at that. And unlike the Patlabor dudes, these fellows DO move against any and all crime, with superior firepower through and through. Unfortunately for the city, they're also all Cowboy Cops... can you say 'Collateral Damage'?
Codename: Sailor V and its better known spin off Sailor Moon have a strange example in Minako Aino, alias Sailor V and Sailor Venus: Minako has a rather vocal dislike for police officers, yet both the manga and anime continuities show her befriending some officers and acting like a Japanese cop when superheroing (and getting mistaken for a Super Cop by a youma during her early days as Sailor V), with the anime even having her as a Super Cop in London for a while as part of her background and the Codename: Sailor V manga ending with Minako accepting to work for the Tokyo Metropolitan Police, meaning she was a Super Cop for the whole run of Sailor Moon without it ever coming up. Shestilldislikes your average Japanese cop.
The Night Watch from Discworld may fit in this trope. Living in a fantasy world, they've got trolls, dwarves, golems, zombies, vampires and werewolves in their ranks; and Carrot Ironfoundersson. Of course, criminals can also belong to any of those supernatural species, which makes The Watch even more impressive because their superpowered members don't necessarily have an edge because of their superpowers.
In the X-Wing Series, Corran Horn discovers that his father, who was part of the Corellian Security Force all his life and got Corran into it too, was the son and the student of a Jedi Knight. Both of them did rely a bit on their Force-Sensitivity. I, Jedi also notes that Corellian Jedi, including Corran's grandfather, tended to work closely with the Corellian Security Force on the tougher cases.
The Aurors (and also the poor forgotten Hit Wizards) of Harry Potter including Aurors Harry Potter and Ron Weasley.. Mad-Eye Moody was a legend amongst them.
Similar to the Sentai above, most Metal Heroes take on this role, as well. In fact the first three, Gavan, Sharivan and Shaider, all share the title Uchuu Keiji, which literally translates to Space Sheriff.
In his first episode of Heroes, cop Matt Parkman is able to telepathically hear a little girl calling for help. In later episodes and seasons his uses his powers to also try to help people, even when he is no longer a cop.
City of Heroes features Blue Steel, Memetic Badass and the only major superhero who works directly for the police force. There are also cops in Power Armour, cops with Psychic Powers and the Awakened division of alien symbiote-infused cops. There's also nothing stopping player character concepts from being cops.
Konoko. Helps that she was secretly bred to be a One Woman Army capable of handling anything the Syndicate could field.
A rare glitch in GTA: Vice City would sometimes render a single random Police Officer invincible, there were also rumors that he could arrest you anywhere, including magically teleporting from the ground onto your helicopter and playing the arrest animation.
Mortal Kombat has Kurtis Stryker. Just a regular guy but packs some ordnance to make up for it. Fights otherworldly monsters so he's going a bit above the line of duty.
Chun-Li of Street Fighter is an Interpol agent, always hot on the trail of M. Bison.