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.
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)
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)
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.