Continuity: Spock lifts his crew mates with the rocket boots. He passes deck numbers 35 through 78 from bottom to top. First off, deck numbers go from top to bottom. The bridge is on deck 1. Second, the Enterprise of that class only had 23 decks!

Continuity: In the scene where Spock, wearing rocket boots, is in the turboshaft carrying Kirk and McCoy on either arm, he fires the booster rockets, thus propelling the three of them up the turboshaft. If you watch carefully, the deck numbers in the background as they rise upwards go as follows: 35, 52, 64, back to 52, 77, 78, and then 78 again! Not content with one goof, Kirk and McCoy can quite clearly be seen with their feet dangling in mid air, not on Spock's boots. They aren't exactly clinging to him either, more like leaning against him. Surely Kirk and McCoy would be unable to hold onto him!

Visible crew/equipment: When Kirk falls at the beginning of the movie and Spock catches him, you can very easily see the tether cord affixed to Kirk's back.