Revealing mistake: When J and K are knocked over by the roach near the end, when they land you can see a square of grass move. Shows that a crash mat is covered with a bit of grass to make the landing easier. (01:19:20)
Continuity mistake: When Edgar/The Bug is drinking water with sugar, the way he holds the glass changes between shots. (00:15:21)
Other mistake: When K is deleting all documents belonging to Edwards (later J), his passport says he was born in Indiana, but his birth certificate says he was born in Manhattan, New York.
Other mistake: The first time the neuraliser is used it shows hours, weeks and years. But at the end of the film the lead character tells J to remember that it's "days, months, years." (00:08:06 - 01:24:52)
Suggested correction: Perhaps there are different models of neuralyzers.
I don't think arguing "perhaps such-and-such" is necessarily strong enough of a reason to correct a mistake. Nothing in the movie indicates they are different models, so it's probably best to assume they're the same.
Continuity mistake: During the written test, Hudson accidentally pierces the page with his pencil. He then reverses the pencil so the eraser is towards the paper. After the cut to Jensen trying to work on his, Hudson's pencil is suddenly corrected with the point towards the paper.
Visible crew/equipment: As K goes to tell Mikey to be quiet, the corner of the sound stage can be seen over his left shoulder.
Continuity mistake: When J holds the baby squid, his fingers protrude more or less depending on the angle. (00:44:40)
Revealing mistake: When the jeweler's ear pops-open, the angle changes and it gets replaced for a prop that looks nothing like the real ear. (00:54:35)
Revealing mistake: When Arquillian is laying flat dead in the morgue, his pulse on his throat is very visible. (00:50:35)
Continuity mistake: In the jewelry store, when J chases the farmer, the boots behind him swap from standing to fallen between frames. (01:03:35)
Continuity mistake: After Laurel is abducted and Jay tries to find her in one of the cabs, Kay brings the LTD around to pick Jay up. The LTD stops beside Jay in a position parallel to the white lines in the road. Jay gets into the LTD and when Kay drives off the LTD has noticeably changed its angle to the white lines even before it begins to move.
Revealing mistake: When James Edwards (later, J) catches the alien in the beginning and has him next to a wall, the alien pulls out a "reverberating carbonizer with mutate capacity" to shoot J. A quick tussle, and the alien drops the gun. When the camera goes to a close up on the gun, watch carefully. You see it quickly fade out of view after it comes to rest, and replaced with a few burning sparkle effects to simulate disintegration.
Continuity mistake: When K and D are talking to Mikey, they are standing in a clearing with no grass on the ground. When Mikey spots the trooper behind him, he turns and starts running towards the trooper, and he is running down a pathway in the grass. When K shoots Mikey and he is blown apart, there is a shot from the trooper's point of view towards K and D, and the pathway that Mikey just ran down is no longer visible and K is no longer standing in a clearing.