Deliberate mistake: When Morpheus is facing the twins' car, the road's not all that wide, and there are concrete barriers both sides of him. After he rolls to the side and slices their car, we see a side shot, quite far back from the road, and the barrier closest to the camera has disappeared (presumably so we get a better view). In a closer view, both barriers are back again. (01:29:25)
Deliberate mistake: In the scene where Neo is sitting with the oracle and talking, there is one quick shot of the fence area where Smith comes in. In spraypaint on the metal fence it has "ONE" written across it diagonally up to the right. When the oracle leaves and Agent Smith comes through the fence entrance the spraypainted "ONE" has disappeared.
Visible crew/equipment: During the scene where Neo is talking to Agent Smith in the park area where Neo was talking to the Oracle, there is a close-up on Agent Smith's face. In his sunglasses you can see a bright white screen to reflect the light onto the faces of the actors. This isn't visible in any non-reflected angles. You can also see the cameraman on the other side. (00:51:15)