Character mistake: Sybok tells Kirk that no one believed the world was round until Columbus proved it. Someone who studied at the Vulcan Academy should know that the Greek philosophers had proved the world was round and calculated its circumference centuries earlier. Columbus never set out to prove the world was round, just to find a shortcut to India.
Character mistake: When they're reviving Lindsey, Bud screams "fight" and slaps her. watch as she closes her eyes on the second slap while she is supposed to be dead.
Character mistake: While pitching his shrinking machine to NASA Szalinski claims that reducing the size of a rocket's payload would result in "staggering" savings in the fuel required at takeoff. However, he makes it clear that his machine works by reducing the space between atoms, meaning that the shrunken object weighs the same as the original. Weight is the critical factor in calculating fuel consumption, not size. Anyone at NASA would pick up on that instantly.
Character mistake: In the movie they grab Genghis Khan in 1209 which would have been correct, he ruled Mongolia from 1206 to 1227. In the presentation at the end they say in 1269 he did this and that, that would have been Kublai Khan's reighn of terror in Mongolia.
Suggested correction: Bill and Ted have proven not to be that smart. Easy for them to get those things confused.
Character mistake: Ray and Egon are giving Oscar the check-up; Ray suggests to run a "Gammill and Pross Infant Acuity Test", which does not exist and is an inside-joke in the screenplay, since Tom Gammill and Max Pross are a duo of comedy writers. Egon agrees and says that they'll finish with an Apgar score, which is a real pediatric term - but it's a test for literal newborns, not an eight months old one like Oscar. (00:16:05)