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.
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.
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.
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.
Character mistake: Captain Holt calls the Pimento case "Operation 225641441636324" which he said he arrived by placing a numerical value to each letter in Pimento and then squaring it. But the perfect squares couldn't match up to Pimento since there's only 6 perfect squares and Pimento has 7 letters (plus the last 2 perfect squares don't fit the pattern of the previous 4). Nor is "225641441636324" a perfect square.
Plot hole: Amy lets Scully place the witch 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 goes back through the window he came from.
Character mistake: When Charles is talking about Terry's kids you can see Stephanie Beatriz (Rosa) trying not to completely crack up next to him.
Character mistake: When Charles gets kidnapped Terry screams out Joe instead of Charles. (00:05:50)
Continuity mistake: When Jake is explaining why he let Richmond go, the left zip of his hoodie goes from folded out to flat between shots and back again a few times. This is especially noticeable when he says "the old me." (00:18:07)
Continuity mistake: The flyers on the top and bottom of the pile Terry is holding keep changing colors.
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.