Plot hole: Someone entered Erin and Nicky's apartment, moved furniture, and disturbed photos. In the morning, Nicky is leaving the apartment to go to school, and she just opens the door without unlocking any of the locks or removing the door chain. It's NYC, and their apartment door isn't secured during the night or day, and ADA Erin Reagan doesn't think to question why the door is unlocked. (00:00:37)
Plot hole: Jamie and his witness are in a patrol car when they are attacked by criminals in a van firing machine guns at them. Jamie (of course) chases them away after a prolonged firefight, and when the van is driving away we see another patrol car also shooting at the fleeing suspects. There didn't appear to be any cars with Jamie, nor did anyone come to his aid during the gunfight, and this other patrol car was facing the wrong direction.
Plot hole: Early in the show, Jamie is in a patrol car escorting a witness when they are attacked by several guys in a van. The side windows of their cruiser seem shatterproof and withstand many bullets, while the rear window is shattered immediately. NYPD cruisers do not have bulletproof windows.
Answer: It's a common cop show plot device. If the police on the show always acted in the most logical way (i.e. not alerting a suspect to their presence) the show wouldn't have the opportunity for an exciting foot chase.