Character mistake: When Gina tries to set up Rosa on a date, Rosa mentions the last person Gina set her up with (Justin) was a dork who rode a Yamaha motorcycle. She says "he might as well have picked me up in a station wagon." Rosa's bike is a Yamaha R1.
Revealing mistake: When Charles shows the picture of the horse and says "look what you get when you search for 'Charles Boyle medal of valor'" he's typed "peanut butter the horse" into his phone.
Continuity mistake: In season 2 episode 11, Jake is shown doing at least 3 pull ups but in this episode he shows he doesn't know how to do one and Terry has to show him.
Continuity mistake: Just before the end credits, Boyle is walking to his desk and he places his jacket over his chair with his bag on top of the coat. The shot changes to Peralta speaking but then the shot goes back onto Boyle and the jacket has disappeared. (00:20:20)
Continuity mistake: When Jake is talking to Gina about Charles' Christmas gift situation, a female cop is walking out from the rest area twice in one scene. (00:01:55)
Continuity mistake: Because the show is filmed in L.A, the lack of snow in December is very noticeable. When the characters are outside there isn't any snow whatsoever, but when Jake, Gina, and Charles are inside the store, the audience can see the heavy snow and blizzard outside from a large window behind them. (00:07:55)
Factual error: When Jake is given a burner phone by Doug Judy, he receives a text message from the 'Pontiac Bandit'. The phone is an old cellphone from the late 90s, but the text message makes an iPhone sound/tone from the late 2000s. He receives another text message again later with the same tone. (00:09:55 - 00:12:50)
Casecation - S6-E12
Factual error: Amy is wearing a NYPD detectives badge even though she is a Sergeant at this point in the show. (00:01:00 - 00:21:20)
Your Honor - S4-E18
Continuity mistake: In the cold open, when Jake throws the door handle at the interrogation room mirror trying to break it, the handle bounces off the glass to his right, but when the shot cuts to in front of him to show his reaction, the handle flies off to his left. (00:01:00)
The Chopper - S2-E22
Revealing mistake: When Terry answers the phone to his wife after he changes his relationship status to "it's complicated", he answers the phone but the phone is still on the answer the call menu.
Revealing mistake: At the start, Terry takes Jake's phone off him while he's in the middle of writing a text. We see the screen and it's on the regular homescreen. Jake then takes it back and immediately carries on typing, not opening up the messaging app first.
Casecation - S6-E12
Continuity mistake: Jake mentions that he's never wanted kids, however he has mentioned loads of times in previous seasons/episodes about wanting to become a Dad.
Revealing mistake: At the end, Gina makes her flame-thrower with cologne and lighter. However, she never lights the lighter and the flames still appear, as if they were just added in digitally.
Other mistake: It is previously explained that Jake cannot grow a moustache, then a few seasons later Jake goes to prison and grows a moustache.
Old School - S1-E8
Visible crew/equipment: While Charles is removing the shoes you can see the reflection of a camera in the closeup shot of the suit.
Old School - S1-E8
Visible crew/equipment: While Charles is removing Scully's shoes, the reflection of a mounted camera can be seen in the close up shot of the bomb squad outfit. (00:00:01)
Visible crew/equipment: When Amy calls the squad in to talk about a Christmas card for Holt, there's tape on the floor by Gina and Terry's feet, showing where they are supposed to stand. (00:05:00)
Continuity mistake: When Adrian stamps "denied" onto Boyle's insurance claim form, there's blank spots where there wasn't enough ink on the stamp (which would happen because he had the stamp in his pocket for a while). But then when he says "denied" and holds up the form, "denied" is all filled in fully. Also, the red border is lighter and the location has shifted down.
Operation: Broken Feather - S1-E15
Continuity mistake: When Gina makes fun of Boyle for wearing a sweater there is a woman behind her talking to cops, when they zoom out it's a man and it changes back when they zoom back in. (00:12:10)
Chasing Amy - S4-E17
Continuity mistake: The position of the money Charles puts on Gina's desk keeps changing through the scene.
Answer: In reality, the Commissioner is a civilian administrator, appointed by the mayor of New York City. He (as of this writing, the role has always been filled by a man; the show gets that part right) is usually someone who has risen through the ranks of the NYPD, but upon becoming Commissioner, is no longer a sworn member of the force (the highest sworn rank is that of Chief). Since Commissioners are appointed, technically anyone can serve in the position, including Deputy Commissioners (who are appointed, in turn, by the Commissioner himself).
Update: On 1 January 2022, Keechant Sewell became the first ever female commissioner of the NYPD.