Continuity mistake: When Agent Smith is interrogating Neo, after Smith has sealed Neo's mouth shut and he is backed into the corner, when the camera cuts back to Smith you can clearly see a reflection in his glasses of Neo still sitting down in the chair.
Visible crew/equipment: When Neo is going to open the door to enter the Oracle's house, you can clearly see a camera on the doorknob. There's a sheet over it painted to look like the wall behind it, with a representation of Morpheus' tie too, because he's blocked by the camera.
Continuity mistake: When Neo is out on the scaffolding and loses his phone, in one shot you see the phone falling towards an empty street, then you cut to Neo, and then back to the falling phone. This time the street is filled with people and a big parade.
Visible crew/equipment: In the scene 'Rooftop Rescue', when Morpheus is hanging from Neo's arm (who is hanging from the helicopter), a safety wire is visible running out of Morpheus' sleeve and into Neo's hand.
Continuity mistake: When Morpheus is fighting the agent in the bathroom of the building they were trapped in, the agent punches through a brick wall, yet next time we see the wall, there is no damage.
Visible crew/equipment: When the 3 agents visit the roof top where Neo saved Trinity from the chopper, you can clearly see the camera crew in the agents' sunglasses.
Continuity mistake: When Neo comes to work he glances at the name of his company. The sign says Metacortex. When Neo is running from the agents there is a sign on the back wall which says Meta Cortechs.
Continuity mistake: When Neo first meets Morpheus, Morpheus extends his right hand with his other hand behind his back. Next shot shows Morpheus with his left hand at his side, and when he withdraws his hand, it is back behind his back. Talk about fast hands!
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.
Continuity mistake: The first time Neo comes out of the Loading Program, and is about to throw up, it shows you a shot from his eyes and the floor is clear of anything, and then it does a shot of him throwing up and he does so on a bunch of wires. The angle changes too - the POV shows all the lines in the grate at about 45 degrees, but when he throws up, they're all horizontal in relation to him.
Other mistake: Whenever Trinity confronts the agent on the rooftop and says, "Dodge this", she points her gun to temple of the agent while his head is turned to her. He glances back, but never fully turns his head. The next scene show him being shot facing Trinity...with a shot in middle of his forehead. Some people think it's to do with the angle he enters the soldier's body to start with, but the angle change is clearly visible before he changes back into the original body.
Continuity mistake: Shadows move about during the rooftop battle - easiest benchmarks are the shadow stretching towards us from the chimney-type thing at the start, then Trinity's stretching horizontally after she throws the knife at the remaining trooper.
Continuity mistake: When Neo is fighting the agent in the subway, he wipes his blood on the back of his hand, but when he puts his fists up the blood is gone.
Continuity mistake: When Mouse is in the room with the brick window, he drops one long box that is assumed to be full of guns. The scene cuts to Tank and then Mouse again. When he gets out the guns, he pushes 2 smaller boxes off the long box which weren't there before.
Other mistake: When Neo comes back from the conversation with the oracle, a part of the cookie, which he has got from the oracle, is bitten off. A few seconds later he wants to eat the cookie and there is no sign of a bite anymore.
Continuity mistake: When Tank comes in to talk to Neo about training in Neo's cell, the strap on Tank's shoulder moves from being on his skin when he's in the doorway to on his shirt in subsequent shots.
Plot hole: When Morpheus and his crew rescued Neo from the Matrix, how was it possible that Neo's real self had his ear pierced?
Other mistake: When Neo is running along while shooting, it switches to a shot of a soldier firing a shotgun. Just before Trinity kicks it out of his hands, he's firing at the same spot over and over. Trinity is behind him, and Neo is moving, so what is he shooting at?
Revealing mistake: When it is Neo's turn to make the Jump he takes a run up, and the camera changes to a brief sideshot of him running across the top of the building with the sky behind him. There is a big fuzzy blobby outline around him which should have been edited out.