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.
Revealing mistake: In the opening credits (the very beginning), Jake is walking and the image behind him is blurred. When it unblurs (right before it the shot changes), the phone number on the building behind him is 213 which is LA, vs. 718 which is Brooklyn.
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)
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.
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 Jake handcuffs himself to Captain Holt, the cuffs around each of their wrists consists of double metal bars connected to a single bar. The direction of the handcuffs changes a few times, going round their wrists the opposite way, which is quite impossible. Then the same thing occurs when Boyle handcuffs himself to Holt. (00:09:35)
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)
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)
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.
Continuity mistake: The scene where Boyle explains the case for Scully and Hitchcock you can see the same two people walking by the room 3 times, coming from right to left in the background. (00:08:30 - 00:09:20)
Continuity mistake: When Jake and Boyle talk about Jake's Dad, a man (grey clothing and holding a cup) in the background (behind Boyle on the left) walks from the back to the front in the same way in both shots. (01:45:00 - 01:55:00)
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.
Character mistake: When Wunch frisks Holt in order to "check for a wire," she appears to forget that her actions are easily visible through the interrogation room's mirror, something a high ranking official in the NYPD would not likely forget. (00:11:41)
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)
Plot hole: Amy lets Scully place the watch in the break-room, but how could she have known that after Captain Holt steals the crown out of the locker, he would go there? And not, for example, go back through the window he came from.
Other mistake: Holt takes the crown out of the back of Jake's cabinet. However this wouldn't be possible as the crown was in the top drawer. This would have a back on it and it wouldn't be accessible by just removing the back of the cabinet.
Continuity mistake: When Terry's wife Sharon walks into the station, she's wearing a purple dress. When she announces her water has broken while in the interrogation room, she's wearing the purple dress. After returning from commercial break, she's still sitting in the exact same spot but now wearing a burgundy tracksuit and lavender top. (Perhaps she changed clothes because her water broke however, it's highly unlikely since she is sitting in the same place and doesn't look like she's left the room).
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)
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.
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.