When Oscar is at the ballpark, the phone rings and he puts his hot dog down. When the man answers the phone, Oscar is typing and the hot dog disappears. When we cut back to Oscar, he is eating it again.
After Oscar throws Felix out of the apartment, the guys are out searching for Felix in Murray's police car. When Vinnie comes out of Felix's old apartment, they are seen driving in police car #578. In the next scene, they are parked and two other officers pull up in car #578, and the guys pull off in car #527. In the next scene, they are back in #578 again.
The apartment's kitchen window is closed when Oscar and Felix enter it together the first time. After Oscar asks Felix to get him a pot from under the sink, the window is open, which is later shut by Oscar.
During the first poker game, Vinnie is dealt a hand and then he rifles through them in full view of the camera (and the other players). The cards are easily visible as two Queens, a Ten, a Nine, and a Four. At the draw, he asks for no cards, and then he wins the hand, announcing he has a Straight.
In Oscar's apt., there is a wall lamp on either side of the front door. Each lamp holds two bulbs, but one of them is missing a bulb. This missing bulb jumps back and forth between the two lamps throughout the opening scenes.
After Felix is chased into Oscar's bedroom, when he first begins to walk to the bed his tie hangs down, then the next shot shows his tie tucked into the top of his pants as he continues his walk to the bed.
When Murray the cop is dropped off at the poker game by an NYPD Radio car, the car's side says "PCT. 231," meaning the 231st Precinct, which does not exist, nor do the precinct numbers get anywhere near that high. The highest numbered precinct in the NYPD (There are seventy-six Precincts today- seventy-NINE in 1968)) is in the Tottenville section of Staten Island, and that's the 123rd Precinct.
Oscar plays a pinball machine in one scene. Pinball was illegal in New York City from 1942 until 1976, but pinball machines did exist. The illegal pinball machine is likely intended to demonstrate the seediness of the pool hall.
When Murray & gang go to look for Felix in an NYPD Radio car, the car's individual number is "578." (Look on driver's side, rear post). Yet when they run into a real, uniformed, on-duty radio car team during search, not only has Murray's radio car switched to "927," but the REAL cops are now driving car "578!"