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)
Character mistake: When Charles gets kidnapped Terry screams out Joe instead of Charles. (00:05:50)
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.
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)
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.
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.