Pet Rat

An unsavory and often sneaky character who is in the employ (off the books, of course) of someone with money and possibly some respected position of authority, and carries out certain dirty deeds on their patron's behalf so the patron can maintain plausible deniability in the matter. This character is often found working (secretly) for the Villain With Good Publicity.

Not to be confused with ACTUAL pet rats, such as Scabbers in Harry Potter.

Film

 * Alistair Harper and his gang are hired in this capacity by the government when the existing police force isn't enough in V for Vendetta.
 * Robert G. Durant and his gang while working for Strack in the 1990 film Darkman.
 * The Brute Squad hired by Prince Humperdink in The Princess Bride. Their job is to clear out the Thieves Quarter.
 * And Vizzini and his crew even more so, really.
 * The Warden does this to the main character in The Shawshank Redemption.

Live Action TV

 * Michael of Burn Notice is treated like this at times. A burned spy, nobody wants to be caught employing him, so all of his work is strictly off the book. Even when  Michael is warned that he'll take the fall if an op goes bad.

Theater

 * In the musical Knickerbocker Holiday, Stuyvesant's first action after his New Era Speech is to secretly hire Tienhoven as his pay-off man. The job will help show that the government can do no wrong by concealing such activities as its sales of guns and liquor to the Indians.
 * Several of William Shakespeare's plays have lower-class characters who are secretly hired by the Villain Protagonist nobleman to murder his rival(s), most notably Macbeth and Richard III.

Video Games

 * Ugarte is the Pet Rat of Ferarri in Quest for Glory II and Quest for Glory V.

Western Animation

 * Zipper, Eric Raymond's go-to henchman in Jem and The Holograms

Real Life

 * Truth in Television. Pretty much every country has a department or three of these guys.
 * In the spy trade the word is "cut-out" according to GURPS Espionage