Other mistake: When Morpheus makes his escape, Agent Smith shoots through the walls with a noticeable time between shots. However when you see the bullets in motion (the 'bullet time' sequence), they're so close together in time that Morpheus doesn't appear to move in the interval between shots. Even if these are supposed to be later shots they could not be fired so close together either by the agents or by the guns used. The first two bullets are some distance behind Morpheus, corresponding to the change of direction of the shots fired by agent Smith to track Morpheus' movement - this also doesn't fit with the shots seen in the 'bullet time' sequence.
Continuity mistake: Near the end, after Neo steals the cell phone, as he's running away, you first see the cell phone owner from the front, waving his right arm and shouting. The next camera angle shows the owner from behind waving his right arm, but he has obviously already been inhabited by the agent (the suit colour's the giveaway). The next cut has the "Agent inhabiting body" sound effect, and he turns around to begin chasing Neo. (01:54:20)
Deliberate mistake: In the rooftop fight scene where Neo unloads 2 pistols at an agent and misses, you can hear the guns firing from behind, followed by 2 clicks suggesting that the clips are empty. Neo's guns are both semi-automatics and would therefore not click when empty as the slides would be locked back after the final shot. The click sounds were obviously added later to indicate to viewers that his clips were empty. (01:46:15)
Continuity mistake: When Morpheus fights Neo in the training program, Neo is pushed backwards and slides along the floor. In the close-up, his head stops level with one of the black stripes on the floor, so that his eyes are looking at it. In the wide shot in which Morpheus jumps in slow motion through the air, Neo's head is past the line so that it is level with his chest, as if he has slid forward further, despite having stopped previously. In the close-up, the line is again level with his eyes, but when we see Morpheus' knees crashing into the floor, Neo has again moved forwards such that his head is no longer level with the line. (00:50:45)
Continuity mistake: When Morpheus shows Neo the agent training simulation, the girl in red turns into an agent. When they pause the simulation you can see the reflection of Neo with a gun pointed at his head in Morpheus' glasses. When you look in one shot there is a visible white line/mark (scenery?) between the gun and Neo, in the next shot it is gone and then right at the end in the final shot it's back again.
Continuity mistake: When cops are in the room with Trinity at the beginning, just before she attacks the first cop, the chair she sat in is really close to the table, yet when she is in the air its a lot further away from the table. Also, the cop holds in his left hand a torch, which disappears when Trinity goes up for the kick. (00:02:50)
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)
Continuity mistake: During the rooftop scene check out the tiling situation. When Neo takes two bullets, he falls to the ground. He lies mainly inside one tile, even the left gun fits in. (And it's on a vertical line) His top hand lies just on the line. Now, watch again when the agent is dead. All of the sudden he's in-between two tiles, and the gun is now on a horizontal line, but not a vertical. (01:46:36)