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)
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.
Sal's Pizza - S1-E9
Continuity mistake: When Jake and Charles are discussing why Sal's Pizza is only seventh in mouth feel, Jake's badge is flipped backwards. When Jake mentions Fire Marshal Boone, it's flipped frontwards again. (00:04:41)
Visible crew/equipment: When Amy calls the squad in to talk about a Christmas card for Holt, there's tape on the floor by Gina and Terry's feet, showing where they are supposed to stand. (00:05:00)
Continuity mistake: When Jake handcuffs himself to Captain Holt, the cuffs around each of their wrists consists of double metal bars connected to a single bar. The direction of the handcuffs changes a few times, going round their wrists the opposite way, which is quite impossible. Then the same thing occurs when Boyle handcuffs himself to Holt. (00:09:35)
Factual error: When Jake is given a burner phone by Doug Judy, he receives a text message from the 'Pontiac Bandit'. The phone is an old cellphone from the late 90s, but the text message makes an iPhone sound/tone from the late 2000s. He receives another text message again later with the same tone. (00:09:55 - 00:12:50)
Operation: Broken Feather - S1-E15
Continuity mistake: When Gina makes fun of Boyle for wearing a sweater there is a woman behind her talking to cops, when they zoom out it's a man and it changes back when they zoom back in. (00:12:10)
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.