From: The Matrix
Crew or equipment visible: When Neo is examining the spoon prior to bending it, a crew member dressed in black - probably the cameraman - is reflected in the spoon.
Crew or equipment visible: When Neo backflips off the train tracks onto the platform, at the end of the flip, a few inches of the wires can be seen either side of his hands.
Continuity: After Trinity crashes through the window, tumbles down the stairs and then points her guns back toward the window, we cut to a close shot of the top of the stairs showing a hanging light fixture swinging back and forth. A few seconds later, during an over-the-shoulder shot from Trinity's perspective, the fixture is perfectly still. The sound and image imply that it did not start swinging until she reached the bottom of the stairs.
Boom mike visible: While showing Neo what has happened to his real world, Morpheus looks to the sky and remarks: "Fate, it seems, is not without a sense of irony." As he looks up, you can see the boom mike reflecting in his sunglasses
Continuity: When Morpheus attacks Smith to save Neo, Morpheus headbutts Smith and knocks his glasses off (camera angle behind Morpheus). But then there is a shot from Smith's POV and his glasses are on again, then the camera shifts to Morpheus' view and they're gone again.
Crew or equipment visible: When Neo reaches to open the door to the Oracle's apartment, the lens of a camera is visible. The camera itself seems to be covered with green cloth to match the color in the hallway, and even has a yellow tie painted on it to match the one Morpheus is wearing, since he is standing behind the camera.
Revealing mistakes: Less than a second before Neo jumps off the building in "the first jump," special effect artifacts are briefly visible around him. (As he's running, side view, against the sky.)
Continuity: When Neo drops the phone, the first time you see the phone falling, there is a parade marching in the street below. Cut immediately back to Neo, the street is empty.
Incorrectly regarded as goofs: The Massive Attack song "Dissolved Girl" playing in Neo's headphones is track 6 on the album, not track 5, and the timing on the counter is off in relation to the actual song. Given Neo's equipment, it could well have been a homemade custom CD.
Revealing mistakes: A security guard calls for help on his radio at the beginning of the lobby shootout. He is wearing knee pads underneath his uniform.
Continuity: In the "dodge this" scene, Trinity's gun changes from a Beretta 92F to a Beretta 84F.
Crew or equipment visible: The "reflection" of Neo's hand on the knob of the door to the Oracle's home shows a hooded camera operator.
From: Matrix Reloaded
Continuity: Bullet holes are already visible on a truck that will be shot at later in the freeway sequence.
Crew or equipment visible: After meeting the Oracle, when Neo is talking to Agent Smith in the courtyard, just before the second Agent Smith arrives: a reflection of a crewmember can be seen in Agent Smiths sunglasses crouching around a corner, possibly with a camera.
Continuity: During the car chase, the twins shoot out most of the windows in the car that Trinity and Morpheus are driving in. But when seen from the inside of the car, the windows are intact and there is no shattered glass to be seen.
Continuity: The cut on Trinity's right arm disappears and reappears throughout the freeway scene.
Continuity: When Neo, Trinity and Morpheus visit Merovingian and are first seated at the long table, Merovingian has a glass of red wine - the amount in the glass changes inconsistently between shots.
Continuity: When Neo is fighting Agent Smith in the playground his sun glasses keep vanishing and reappearing.
From: Matrix Revolutions
Continuity: When Bane is talking to Neo while holding the knife to Trinity's throat, the position of the knife changes, and blood appears and disappears on her throat.
Continuity: The last time Neo goes to see the Oracle, she puts her bracelet on twice.
Continuity: When saying goodbye to Morpheus, Neo's and Trinity's brain jacks disappear and reappear between shots.
Continuity: When Neo is talking to the Oracle the magnetic letters on the refrigerator behind him disappear then reappear.