Bad Boys: Detectives Lowery and Burnett of the Miami PD. They pissed off Da Chief so badly that he asked them if they called each other in the morning and discussed ways to make his life miserable. In one scene in the first movie, Lowery pulls out two guns and puts them to the head of an innocent person to get him to tell them who might be capable of cutting the heroin for the bad guys.
Axel Foley from the Beverly Hills Cop movie series. In the first movie alone he takes a truckload of cigarettes without authorization and wrecks it, is chewed out by Da Chief and warned that if he commits any more offenses he'll be fired. Then a friend of his is murdered and he's told to stay away from the case and again warned he'll be fired if he doesn't. Of course he ignores the warning. In the sequel, Rosewood becomes one as well.
Deconstructed in Blitz, where Jason Statham's character, Sergeant Brant, has severe anger issues that lead him to excessively brutalize criminals. This is shown in the opening scene, where he beats the crap out of a trio of car thieves with a hurling stick ("An Irish sport, cross between hockey and murder.") He is repeatedly chewed out both in the media and in the police station for his antics. This tendency resulted in him thoroughly beating down Weiss, the movie's Serial Killer antagonist, about a year before the movie began, resulting in him being hospitalized. Weiss, already a bad apple before Brant beat the everloving crap out of him, was apparently driven off the deep end by the experience and snapped, turning into a cop-killing lunatic.
David from Bon Cop, Bad Cop is a Québecois Canadian take on this trope. His reaction to Martin telling him how much rules he breaks: "J'm'en câlisse" ("I don't give a shit").
The Trope Maker is probably the Steve McQueen movie Bullitt, which is also considered to have invented the Car Chase. Due to the Unbuilt Trope, however, the cop turns out to be an an absolute screwup, essentially ruining any chances of catching the real bad guys.
Bullitt was told to guard the wrong man, uses that man's shooting to trap his killers, embarrasses the DA who sent him to guard the wrong man, tracks the right man down to the airport and prevents his escape from justice with the millions of dollars he stole. The only thing that Bullitt screwed up is when he went to arrest the right man, he couldn't prevent the suspect from fleeing back into the airport and iniating a gunfight with security and Delgado. That leads to Johnny taking some lead instead of testifying.
Detective Rainer in The Car: Road to Revenge. The first scene of the movie has his commanding officer ordering him not to engage with Tinkerman until backup arrives. Rainer immediately ignores the order.
Charles Bronson played a lot of these cops back in the day, who circumstances would force to take the law into his own hands and become a Vigilante Man delivering violent justice, such as in the movie Kinjite: Forbidden Subjects, where he's forced to do just that against a ruthless pimp running a child prostitution ring that has victimized the daughter of a Japanese businessman.
Cobra - Marion Cobretti (Sylvester Stallone). The weirdest thing about Cobretti is that he never really uses excessive force. Everyone he kills in the film tried to kill him first. However, since Cobra is So Bad, It's Good, his superiors act like he is anyway, making it look like self-defense is against police procedure.
Coogan's Bluff, a proto-Dirty Harry also directed by Don Siegel and starring Clint Eastwood. Unlike its successor it plays the trope completely straight: Coogan is an Arizona Sheriff's deputy who's sent to New York to extradite a murderer, engaging in all kinds of Fish out of Water antics in the big city before he brings the killer to justice.
Cop: James Woods' character in this movie, based on a James Ellroy script, is less of a "badass, action-y" take on the trope and more of a Defective Detective. He's antisocial, a bit corrupt, repeatedly ignores police procedures while investigating a serial killer case, gets himself implicated in theft (for stealing evidence from a person of interest) and murder (for shooting a suspect after beating up and interrogating him). By the end, he's about to be thrown out of the police force by Da Chief and with nothing to lose, dispatches the killer he was chasing in a Vigilante Execution.
Dark Blue: Deconstructed. Veteran LAPD cop Eldon Perry flouts all the rules because he figures that to catch bad guys his town was "built on bullets" and doesn't object to framing and flat-out murdering suspects to quickly close a case. This ends up gradually destroying his life and reputation and gets his partner killed.
Even lampshaded in the exchange with one of the bad guys in the first movie:
Tony: You won't kill me.
McClane: Oh yeah? Why not?
Tony: Because, you are a police man. There are rules for police men.
McClane: Yeah? That's what my captain keeps telling me.
Dirty Harry: Harry is a classic example. The films also make fun of this trope and show people reacting somewhat more realistically to it. In these movies, unlike the cop shows on TV, people actually notice and complain about how Harry is always running up the body count, how he keeps totaling (very expensive) police cars, and the way his partners keep getting shot.
Tina from the Fantastic Beasts series got demoted from being an Auror before the first film for attacking Mary-Lou Barebone for abusing her son, Creedence, while on official duty.
Joe Don Baker's character in the film Final Justice (as seen on MST3K) is a figurative and literal example, as a Texas lawman who doesn't let little things like international boundaries get in the way of gunning down criminals.
Nicholas Angel in Hot Fuzz is an inversion - he plays completely by the rules and so perfectly that he gets assigned to a sleepy little town because he makes all the other cops in London look incompetent. But later plays the trope up in order to bring in the real criminals.
Unfortunately even in London standards his excessive paperwork of everything makes him a Cowboy Cop of protocol, most cops will simply file a generic report for minor misdemeanors or even just brush it off as a mischief charge but he goes to the point where he does everything from the police lineup to precise details of how much alcohol a teen had taken. In short, Nick dispenses red tape instead of bullets.
I Come in Peace: Jack Caine prefers to play by his own rules, but this does get him into a lot of trouble at the start. The only reason he isn't booted off the force is because the FBI agents pull some strings.
Hard Boiled - Inspector Tequila. Also in the related game Stranglehold. In Hard Boiled, his partner is killed and after shooting the killer in cold blood, he's constantly going against his superiors orders to take down all the triads. Stranglehold is basically the same (Even with a Turn in Your Badge) with an added kidnapped loved ones.
The Killer - Inspector Lee. Aside from some disagreement with his superiors he also spends the third act helping an assassin take on an army of Mooks.
Arnold Schwarzenegger spoofs this trope in the film Last Action Hero. In it he plays Jack Slater a trigger happy cop who meets all the requirement for this trope as well as going against a few. At one point he explains to Danny (a little boy who becomes his partner) and later his Captain that he doesn't want to partake in action sequences and fancy heroics.
The film Little Sweetheart contains a mild example. It's not generally legal to shoot a wanted bank robber and accused (but not charged or tried, or guilty) child murder in the gut with a shotgun when he's unarmed but running.
Mac Stern from the buddy-cop film Loose Cannons, fills the role of Cowboy Cop with his toughness and unconventional police methods.
John Wayne started playing these in the 70s. He turned down the role of Dirty Harry and later regretted it. He first starred in Mc Q in 1974 about a Seattle police detective. A year later he made Brannigan, about a Chicago cop that's a Fish out of Water in England. He retired from westerns with The Shootist but would have continued making police movies if he hadn't died of cancer in 1979.
Played for laughs in The Other Guys with Danson and Highsmith. They manage to cause $12 mil in property damage bringing in a bunch of guys in possession of half a pound of marijuana.
Highsmith: You have the right to remain silent, but I wanna hear ya scream!
Predator 2. LAPD Lieutenant Harrigan is ordered to not interfere with a Federal investigation of multiple murders (committed by the Predator). He defies the order and his Captain threatens to fire him if he doesn't stop.
Roy from R.I.P.D. is a literal example of this trope. It's also mentioned that his antics have kept him in the beat for way longer than a century.
Rush Hour - Both Carter and Lee. Especially Carter — he causes massive property damage, uses highly questionable investigation techniques, and does not bother hiding the fact he smokes weed.
In Running Scared (1986), detectives Hughes and Costanzo: arrange for a drug lord's car to be illegally towed and later steal it from the impound yard, steal a drug dealer's money to get him in trouble with his boss, break into homes without a search warrant, and are hinted to have violated other suspects' rights before the movie starts.
The Dolph Lundgren/Brandon Lee Buddy Cop MovieShowdown in Little Tokyo. After Kenner reveals to Murata that Yoshida was his parents' killer and it becomes a personal vendetta, Murata agrees to screw protocol and go after the Yakuza guns blazing. From there on they slaughter the bad guys en masse rather than arrest them and end up burning the bad guy alive in public. Lampshaded by Murata when he feels uneasy explaining what they did to the cops and he and Kenner run away and at the end wonders how they're going to explain everything in their report.
Harley Stone (Rutger Hauer) in Split Second is a complete rundown of the trope: He's a multiple gun-toting hardass who treats everyone like crap, shoots first and asks questions later, pisses of Da Chief and treats his by-the-book partner like a nuisance and gets the job done with superior firepower.
Spock has evolved into this by the time of Star Trek (2009). In a scene between Nimoy and his past self (Zachary Quinto), he encourages Young Spock to think outside the box and bend rules once in a while. This is inverse parallel development to Kirk, who is a thrill-seeking rule-ignoring Casanova Wannabe who causes chaos everywhere he goes; Old Spock tells him to take it down a notch.
It is mentioned that slaves have explosive transmitters implanted inside and and if they make any attempt to escape, their owner detonates it.
Reconstructed in Street Kings as Biggs recognises that Ludlow's methods, while questionable, are sometimes necessary.
Ludlow: If you're going to do something, do it now. I know you want my scalp nailed to your wall, but then who's going to go in where the law won't? You, Captain? You? You gonna clean up the needles and baby parts? No. You need me and my company of men. You hate me, but you need me.
Biggs: Ludlow, maybe you're right. Maybe we do need you. But goddamn if you don't need me, son.
At the end of the movie:
Biggs: One day, you will pass the chief in the hall and he will give you a nod. And you will know why. Because you were right, Tom. We do need you.
Alonzo Harris from Training Day is one of the rare truly villainous examples. He's long become more extreme than even the gangsters he fights, but the reason he's kept around by his superiors (the three wise men) despite his personal corruption is that he catches a lot of bad guys. Alonzo himself claims he is only going after the big fish in the drug trade; he has 38 cases pending trial, 63 active investigations, 350 log cases he has yet to clear, and is supervising five other officers besides Hoyt.
Where the Sidewalk Ends (1950) provides an early deconstruction. Detective Mark Dixon (Dana Andrews) flouts authority and beats suspects, but it all catches up to him when he accidentally kills an innocent man. He was legitimately defending himself, but his superiors have been getting increasingly frustrated with his brutality and might not believe him, forcing him to go to ever-increasing lengths to conceal what he did.
The main character of Yakuza Graveyard at first comes off like a parody of this trope. He'll brawl with anybody, he beats up suspects and threatens to railroad them, he drinks, and repeatedly drives his fist into his palm when frustrated, even if he's talking to his superiors. But it becomes increasingly clear that he's a very sick man, whose constant anger is probably the product of all the beatings he took as a kid (because he grew up in Manchuria, despite being pureblooded Japanese). Gets even further deconstructed as he meets and eventually bonds with a Yakuza member over their shared racial struggles, straightforwardness, and love of brawling.
The titular character of the film Osmosis Jones is a rebelious and to the point white cell, which is the body's version of the police force. He has a clash of personality with his new partner Drixanol Koldrelief (a by-the-book cold remedy pill).
Played straight to the point of absurdity in Michael Cimino's The Year of the Dragon, in which Police Captain Stanley White, who hates Asians indiscriminately (he at one point confuses Chinese and Vietnamese), throws away the book in taking on the Chinatown gangs. He outdoes himself after they kill his wife. Things ascend into self-parody in a brief epilogue in which he attempts to arrest several thousand marchers celebrating a Chinese holiday all by his lonesome. Heaven's Gate was not Cimino's only crime.
The 1973 film The Seven-Ups - best remembered for the chase sequence featuring Bill Hickman of Bullitt fame - features an entire unit of cowboy cops in the NYPD. The unit resorts to below-the-belt tactics to pursue suspects who commit crimes punishable by sentences of seven years and up - hence the title.
Judy Hopps in Zootopia earns the distinction of being a Cowboy Cop who - mostly - follows the rules to the letter, making use of ample Loophole Abuse to aid her in her pursuit of justice. On her second day on the force she abandons her post to chase after a thief, without even calling in for backup, and continues the pursuit into the fragile Little Rodentia district, endangering its citizens. She later bluffs Nick into helping with the case by threatening him with a tax evasion arrest (being a federal offense it's outside her jurisdiction) and manufactures an excuse to enter a property without a warrant.
The Hitman: Chuck Norris's character is actually a cop who's working undercover as a hitman for the mafia, but still fits the 'renegade cop' mold, usually just straight up executing people instead of arresting them. He even blows up his crooked partner at the end in front of the rest of the department.
In Mystery Road, Indigenous detective Jay Swan is on the outer with most of white colleagues and is told multiple times not to work too hard on his investigation. He doesn't listen. In the sequel Goldstone, his actions are even more extreme as he is acting on his own without direct supervision.
The Highwaymen: Hamer uses methods that would be regarded as police brutality by a metropolitan officer, but Texas Rangers in the old days really did have this reputation. In the opening, the then-Governor of Texas, "Ma" Ferguson, even balks at the idea of sending a "cowboy" to hunt down criminals.