In interrogation, Trucho puts the ice pack to his eye twice with no time to put it down between.
When Beckett is drinking coffee in the break room talking to Castle about how she let her mother down, in the close ups, she's holding the mug by the handle, but in the wide shots, she is holding it by the bottom.
When Montgomery calls Beckett into his office and offers her his "poor man's painkiller," you can see her open the flask and take a drink as the camera focuses past her on Montgomery. The camera then immediately changes angles to view her from the front, and she can be seen opening the flask again and drinking from it, though she had no time to lower her hand or close the flask.
When Castle brings Beckett food to her desk, and is listing off all the different kinds of food, he says "Italian" while holding the kind of carton that Asian food usually comes in. The next cuisine he lists is "Thai", as he reaches back into the bag. When the scene cuts back to him again, there is a container of Italian food sitting next to the Asian food carton.
Beckett cuts open a package of drugs and puts it to her tongue to find out what it is. No cop in their right mind would ever do this as it could be poisonous and is against most, if not all, police policies and procedures. That's why most police carry or have access to drug testing kits.
When the detectives examine Jacky's shotgun, one [pointlessly] notes that the shotgun slugs are "hollow point". Then Beckett notes that said slugs would drop a Kodiak in Kevlar®. Actually, most Level IIIa body armor stops most 12 gauge shotgun slug rounds (even though the wearer would be hurting).
It is mentioned repeatedly that no one knows what "Rathborne" looks like. But Coonan has already said he was in the service with Rathborne; U.S. Armed Services records include photos, which the NYPD can request.