Revealing mistake: During the fight scene between Neo and the agent in the subway station, after Neo incapacitates the agent and flips out of the way of the oncoming train, you can see where the wire harness connected to his back. You can also see one of the wires visible by his wrist. (01:57:38)
Revealing mistake: When Trinity's being chased by the cops in the first scene and she makes her first jump from one roof to another when being chased by the cops. One of the cops that is chasing her doesn't quite make a complete jump to the other roof. During his failed attempt when he hits the lip of the roof of the adjacent building it makes the whole building shake. (00:04:00)
Revealing mistake: When Neo shoots the guy with the newspaper at the beginning of the lobby scene it is obviously a prop. It is very thick paper stuck together almost like cardboard because the individual pages are not seen where it rips. Also it would be near impossible to split a newspaper like that even though it does take a few bullets. (01:37:15)
Revealing mistake: When Neo and Agent Smith are fighting on the subway tracks, Neo leaps up to avoid being hit by the train. While the shot is in slow motion, you can actually see the wire supporting Keanu Reeves (just look at Neo's hands grabbing them). The wire becomes momentarily visible right when he is reaching the highest point of this jump. (01:58:15)
Revealing mistake: When Agent Smith is fighting with Neo in the subway, Neo's back is against a pillar. The area of the pillar which Agent Smith punches away with the first punch is visible before Neo ducks, and then he punches it away. You have to look carefully, because it only appears for a very short period of time. (01:50:35)
Revealing mistake: When Neo stops the bullets being shot at him in mid air and they fall, you can see that there are not grooves in the bullets. As a bullet passes through a barrel, it picks up the impressions of the lands and grooves of the barrel. Since the Matrix is a simulation of our world, including the involved physics, the lack of rifling grooves is an oversight by the special effects team.