Continuity: John Bigboote's glasses disappear and reappear between shots when he holds Professor Ikeda in a head-lock.
Revealing mistakes: When the two hunters are poking at the thermopod up in the tree, a shower of sparks falls when they hit the pod. You can see the power-cable they used to produce these sparks fall away as the pod slides out of the tree.
Continuity: During the closing credit sequence with Buackaroo Banzai and the gang walking in the aqueduct, Perfect Tommy is first seen wearing a white jacket and black pants. As they take their final turn in front of the "Buckaroo Banzai" graffiti on the wall, Perfect Tommy is wearing a completely different outfit - gray jacket and red pants.
Revealing mistakes: When Dr Lizardo is pulled out of the wall after his failed attempt to access the 8th Dimension, the hole in the wall in which he is placed is visible.
Continuity: Dr Hikita marks the wall through which Dr Lizardo tries to access the 8th Dimension with a chalk "X". After Dr Lizardo bashes his assistants' heads together, the "X" is missing from the wall.
Continuity: When Secretary of Defense McKinley tries to bribe Blue Blazer Irregular Scooter Lindley with a 20 dollar bill, he is distracted by Perfect Tommy and Scooter snatches it out of his hand. In the wide shot following that close up you can still see the 20 dollar bill in McKinley's right hand.
Continuity: Throughout several points in the movie, after Buckaroo has been "zapped" by the space aliens and supposedly cannot be touched without shocking the toucher, he touches or is touched by several people without them being shocked.
Continuity: During the battle at the Yoyodyne plant the shot of one of the aliens being shot and falling from a catwalk (elevated walkway) is repeated.
Factual errors: On one of the hunters finding an issue of the Buckaroo Banzai comic book, he notes that "it's the latest issue." However, the comic is clearly marked with the issue number - 1. The first issue of a series cannot logically be the latest issue.
Continuity: After Buckaroo briefs the president on the situation, the president replies "I don't know what to say. Lectroids, Planet 10, nuclear extortion, a girl named John?" However, Buckaroo didn't mention John Emdall during his briefing and the president doesn't watch the video until later.
Factual errors: While all of the Yoyodyne employees applied for Social Security cards on 11/01/1938, the "Halloween" broadcast of Orson Welles' Mercury Players occurred on Sunday, 10/30/1938. New Jersey (Jeff Goldblum) incorrectly cites the air date of the broadcast as Halloween (which would have been Monday, 10/31/1938).
Plot holes: New Jersey explains that the Lectroids emit a chemical which humans breathe, and as a result they make us see them as human. However, at that same time, the President is seeing John Parker on television as a human, not a Lectroid, even though he cannot possibly be inhaling the chemical hundreds of miles away.