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: 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.
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: When the lights go off after attaching the belt to the ceiling, Holt is interrupted before he finishes the word "winner", however in the flashback he says "winner of the 5th annual-" before being interrupted.
Other mistake: When Amy reaches down to get the key out of her shoe she literally only bends slightly. There is no way that she could've reached all the way down to her foot, or retrieve the key in the seconds before she gets it. (00:14:30)
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.