During Buckaroo's guitar solo at the night club, he steps over next to the two saxophone players. As they sway together, the camera cuts to a tighter shot of the two sax players. On the left Buckaroo swings his guitar around behind his back (the guitar body visible over his shoulder) getting ready to grab his trumpet, while his solo riffs continue for another five seconds.
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.
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.
When the Lectroids in the Yoyodyne van do a 180 to head back to Route 3, their headlights turn on. In the very next scene, when they run Buckeroo's Harley off the road, they are off again. [about 35 minutes in]
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.
When Banzai is in the rocket car, and plugs the module in, you can hear them reading out his location as N28º 36' 30.32", W80º 36' 14.8". This is the location of the Saturn causeway at Kennedy Space Center. NOT the middle of a desert, with a convenient mountain to fly through.
During the jet-car test, after they have launched, radio traffic is heard saying "APU start is go". This refers to an auxiliary power unit, a small engine used to run systems and start main engines. This shouldn't be needed after launch. It is also heard prior to launch, and makes much more sense then.
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).
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.
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.
At the end of the closing credits, when we see "Buckaroo Bonzai" painted on the concrete wall, you can faintly see erasures on the wall, indicating that they needed several attempts to get the text right.