Continuity mistake: A criminal steals Fish's gun and holds the squad hostage. He has a handcuff on one wrist with the other one hanging. The hanging cuff goes back and forth between being open and being closed throughout the scene.
Revealing mistake: The actor that played Barney Miller, Hal Linden, was left handed and did everything left handed. However, they always put his gun on his right hip with the handle facing the right. A lefty cannot conveniently pull the gun this way.
Suggested correction: Hal Linden writes with his left hand, however he uses his right for many things. He ate with his right hand (in Graft), and in the episode where Fish thinks he needs to retire, Barney pulls his gun with his right hand.
Uniform Day - S6-E17
Factual error: It is Uniform Day where they all have to wear their uniforms. The badges they all wear are those of patrolmen. The problem is that they are detectives and the NYPD has detective badges, which are distinctly different looking.
Character mistake: Levitt is often seen wearing a cap device as a shield (badge). This is a big no in the NYPD.
Grand Hotel - S2-E7
Factual error: A hotel worker citizens arrests a guy for statutory rape in his hotel of his girlfriend who is 17. The police talk about charging him but not wanting to. 17 is legal age of consent in New York so there would be no statutory rape.