When Skip McCoy is smoking in Captain Tiger's office, at one point he scratches his cheek with his right hand while simultaneously holding a cigarette. However, on the next immediate cuts, in rapid-fire fashion, his right hand is hanging down out of view and then it quickly goes back to scratching his cheek.
(at around 35 mins) When Captain Dan Tiger has a conversation with Skip McCoy, a pillow appears to be resting over the hammock's ropes when the camera is shooting from behind of Skip, while the pillow appears to be laying on the peace of fabric when the camera is shooting from in front of him.
When Skip and Joey fight in the subway station, they are fighting in a cleared area near the ticket office and then the next two shots show them confined by exit railings near the newspaper stand and then the next shot cuts back to the open area.
Joey tells Candy that the stolen film is of "a new patent for a chemical formula" and "we can't get another copy of it". Evidently it's a stolen copy of a classified patent (the theft of which the government is obviously aware of, using the theft to identify and round up a spy ring responsible for it). Thousands of patents are classified in the United States. The 1951 Invention Secrecy Act provided the legal framework for doing so, allowing the government to grant patents but maintain them as classified under grounds of national security.
When Skip gets off a subway train at the 33rd street station, he is getting off of an IND line R-1 train. There is no 33rd street station on any IND line. The only 33rd street station is on the Lexington Ave Line (today known as the #6 train). The Lexington Ave line is a branch of the IRT line and did not use the R-1 cars. They used the Low V cars. An R-1 car was too wide and would not fit on to the IRT tracks.
When Joey leaves the basement after assaulting the police officer, he walks out of the shot, then a cat is obviously thrown into the frame.
When McCoy and Joey are fighting they easily knock over a gumball machine. However, in a subway station, the machine would have been bolted down, so it is obvious that it is a prop.
At one point, Candy refers to an address in New York City on Houston Street (pronouncing it like John Huston or the Texas city). In reality, the street is pronounced "House-ton" - something known to every New Yorker.