Max's teacher is reading to her class a story from Dr. Seuss' "Green Eggs and Ham". Jim Carrey later played title characters in Seuss-based movies How the Grinch Stole Christmas (2000) and Horton Hears a Who! (2008).
Randall 'Tex' Cobb, has a cameo in this movie talking about returning a jacket he had borrowed from Fletcher. He also had a whole character in the movie but the scene was later deleted.
When Fletcher is chasing the 747 down the runway, the plane is almost ready to take off. This means it is still taxiing to the end of the runway, giving Fletcher more than enough time to keep up with it and stop it. The call saying they are ready to take off comes right before Fletcher throws the shoe, which would've put it at that impossible speed.
When pulled over by the motorcycle cop Fletcher admits to unpaid parking tickets and opens the glove box of the car and the tickets all fall out. This is impossible since the Mercedes SL series including Fletcher's SL500 never were fitted with glove boxes because of interior space limitations. The dashboard shown in this scene is from a difference series of Mercedes cars.
Fletcher: Where would Tina Turner be right now if she'd rolled over and said, "Hit me again, Ike, and put some stank on it!"? Rollin' on the river, that's where she'd be. But she's beyond Thunderdome, because she decided to send a message?