Continuity mistake: When 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: In the elevator shaft/rocket boot sequence, the 'arm' of the lift device that is actually holding Spock casts a strong shadow on the wall to our right of Spock, indicating the arm goes from Spock's back to the black vertical slot in the right wall. It is most visible as Spock first lowers into view next to Kirk. (01:02:40)