Character mistake: Jake and Rosa are looking around in The Apartment for months. 14 CIS workers are analyzing The Apartment. All of them are looking for a killer who never left The Apartment. Still, in multiple months no-one thinks about checking the person-sized vents. Seems pretty unlikely. Vents are literally the first thing anybody thinks about when people try to disappear. And a vent of this size is a dead giveaway that someone for sure would have checked.
Revealing mistake: When the motor home explodes, the flames from the explosion are CGI. It doesn't look very realistic at all.
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).
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.