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.
Episode #1.3 - S1-E3
Plot hole: The killer is meant to create the perfect crime, but the angle of the shot would make obvious to any forensics examiner that it's a suicide. There is practically no possibility for a person supposedly positioned across a long dining table to kill someone else with a shot under the chin. (00:55:50)
Plot hole: To identify the owner of a severed leg, Pride asks Loretta if she can guesstimate the body's general build by the foot's boot size. A much simpler way would be to swab for blood where the leg was severed and run it through the military DNA database.
Plot hole: The mysterious neurosurgeon Dr. Quinn was mentioned by the fake Myles Taggart under the pretence that his presence was announced by the local newspaper. It turns out that his invitation by the university was a lie - presumably no newspaper story existed at all then, or the university would have acted on that. Taggart is exposed as a fraud, Quinn's link to him is so direct that he was even the one who invited him (instead of goading the Colonel into inviting him like it was his own idea) and yet the Colonel continues to trust Quinn blindly - he even has the whole night and day to think about this simple fact, nothing was decided on the spur of the moment.