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.
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.
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 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.
Mr. Santiago - S4-E7
Revealing mistake: When Jake is typing on his computer just before Terry talks to him about meeting Amy's dad for the first time, his computer is and remains on the desktop doing nothing. (00:02:05)
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.
Revealing mistake: When the motor home explodes, the flames from the explosion are CGI. It doesn't look very realistic at all.
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.