When Eric drives up to the dock to fight, he is still wearing his vest. When he gets out of the car, the vest is gone.
When Eric gets out of his car at the docks and is approaching Ned to confront him, watch his car in the background. A shadow falls over it and it looks like somebody is splashing something onto it. A moment later, Colonel Dugan and some Alpha Elite come out of hiding and one of them runs over to the car and starts splashing gasoline onto it. Whoever was splashing something onto the car a split second before has completely disappeared.
After the scene in which Miyagi compares the 'big rock' to Japan and asks Julie to perform a front kick from it, her pants change color from black to blue in the following scene.
At the beginning of the Senior Prom scene, an overhead view of the gym is shown as the dance is in progress. Julie and Eric can be clearly seen dancing in the middle of the dance floor. But the next scene shows the two of them arriving at the dance; they couldn't have been on the dance floor before they got there.
When Julie wakes up at the monastery, her hair is messy when she looks out the window, she immediately runs to meet with Mr. Miyagi. When she gets there, her hair is all neat.
Once a ball is in the gutter, even if it comes out to strike a pin or pins, it is still scored as a zero. If it is the 1st ball, the pins that have fallen are reset in their positions.
Miyagi is an Okinawan Goju-master (Naha-te based system) the form/kata Julie is practicing is based on the Naihanchi (or Tekki forms) from Shuri based karate systems . She would not practice these as her grandfather learned Goju from Miyagi and these forms do not exist in that system. Most Naha-te practitioners actually showed resentment towards the Shuri system as it represented the 'wealthy' and 'nobility' of Japan, while Naha system was for the working class man.
Mr. Miyagi is a CMOH (congressional medal of honor) recipient. When he gets out of the car in the first scene, the escorting soldiers do not render salutes to him as all CMOH recipients are entitled to.
In the credits, the character Buddhist Monk is misspelled as Buddist Monk.
When the monk bowls a gutter ball (which jumps out to make a strike at the last second) the 10 pin begins falling before the ball contacts any of the pins.
During Julie's fight with Ned that she doesn't actually touch him when he goes down.
When Angel flies free for first time, she is attacked by a wild bird protecting it's nest.
Abbot bowls his ball down the gutter and it hops out, and as he turns back to the crowd, you can see just over his shoulder the mechanism which caused the ball to jump is still extended in the gutter.
In the opening scene, when the camera passes the Fanfare section, a stage light and some crew are reflected (upside down though) in some of the larger horns and tubas.
Angel is a Harris Hawk, a species native to the desert Southwestern US, not the Northeast.
Miyagi tells Julie that her grandfather saved his life and in return he taught him Karate, however in The Karate Kid (1984), Miyagi tells Daniel that he has never taught Karate to anyone before.
Ned falsely accuses Julie of smoking on school grounds and reports it to Dugan. If this were true, Dugan would have smelled the tobacco smoke.
Julie's grandmother, in a moment of anger, calls her "Susan". This implies that Julie is her maternal granddaughter, her daughter's daughter. Julie is her paternal granddaughter, the daughter of her son.