When Jake and Charles are talking in the middle of the bullpen after returning from Sal's the first time, in some shots Jake's badge is flipped backwards.
Fire Marshall Boone ignites a puddle of gasoline by throwing a lit cigarette at it. Cigarettes are not capable of igniting gasoline.
After crushing the Magic 8 Ball in his bare hand, Terry reads off the phrase, "Answer uncertain. Try again." That is not a phrase that appears in a Magic 8 Ball.
Captain Holt wears his Commendation breast bar above his World Trade Center Breast Bar. When the WTC bar is worn, it is worn above all other medals, but below the American Flag bar.
Jake and Charles are somehow able to put a kiddie pool full of pizza sauce at the base of the fire pole in Marshall Boone's firehouse. Even if they could bring in a pool and fill it with sauce without anyone noticing, the fire pole is anchored into the ground. The only way to get a pool underneath it would be to cut the pool but the pool seen here was complete intact.