Suspiciously Idle Officers

In any film or show in which Dirty Cops are secretly in the employ of someone not a member of the police force (e.g., The Mafia or a Corrupt Corporate Executive), they apparently have no actual, official duties, and can therefore spend 100% of their time doing the bidding of their secret employer. No mention is ever made of supervisors, time cards, patrol schedules, or other cases upon which the officers should be working.

Examples of Suspiciously Idle Officers include:

Film

This was a very common trope in Bollywood films from the 1970s to the early 1990s. Most crime bosses had a few cops on their payroll who could be seen hanging around the villain's lair in most scenes without actually doing anything coplike.
In the thriller Confidence, two members of Con Man Jake Vig's gang are actual members of the LAPD, whose role it is to bust in on the scene of the con and force the victim to leave in such a rush that he abandons his money. Eventually, it is revealed that Gunther Bhutan, the federal agent who has seemingly come to town to bust Jake, is actually on his payroll, too. One wonders how he justified the trip to his supervisors.
In the Mob comedy The Freshman, the young student played by Matthew Broderick is unwittingly drawn into the schemes of a mobster played by Marlon Brando, and soon finds himself pursued by two agents of "Department of Justice, Fish and Wildlife Division." Eventually, it is revealed that these agents are actually in the pay of a rival Mob family, and plan on killing both the student and the gangster and stealing the proceeds of their crimes.
The DEA agents in Léon: The Professional are hard-pressed to explain the brutal gun fight that stirred the plot into action, but nobody ever questions why they were there in the first place, and they are never seen pursuing actual police work throughout the film.
Detective Liebowitcz in Sin City seems to exist purely to punch Bruce Willis in the face. His comic counterpart would later be seen walking around, taking orders from a guy who controls a guild for assassins.
- This trope is inverted later in the same movie where cops actually arrive 'too quickly to the scene of the crime in Marv's story, clueing him in that they work for the Big Bad and are trying to frame him. He's right, too.

Live-Action TV

Damages is quite enamored of this trope. In the first season, it is revealed in the finale that Arthur Frobisher's chief goon, who has been handling various bits of Dirty Business ranging from surveillance to contract murder, and who is responsible for the murder of the protagonist's fiancee, is actually a highly-decorated police detective. His efforts to keep his criminal activities secret form a major subplot in the second season. Meanwhile, in the main plot of the second season, a pair of FBI agents who are investigating Patty Hewes' law firm are revealed to actually be in league with one of the targets of her latest case. In this instance, these agents do have a supervisor looking over their shoulder. It's simply that he's in on the conspiracy, too.
In the show Life, which actually features as its Myth Arc an investigation into deep-rooted corruption inside the LAPD, Roman Nevikov, a member of The Mafiya and a Dragon with an Agenda to the members of the conspiracy, is revealed to have a seemingly endless supply of FBI agents on his payroll, enabling him to set up entire fake operations that are actually covers for his own criminal dealings, with the Bureau higher-ups apparently none the wiser.

Webcomics

In Dominic Deegan, before Szark Sturtz's duel with Scarlatti, a batallion of Callanian knights arrives as part of Serk Brakkis's entourage, to make sure the duel is fought in all fairness. They're on Serk's pay roll, and intend to make sure Scarlatti (his proxy in the duel) wins.