Continuity mistake: When Savvik and David first encounter Spock's photon tube "casket", they open it to reveal the burial robe rather nicely folded with the Vulcan lettering shown. When David reaches in to retrieve it, the robe is very disheveled, lying more lengthwise, and the lettering not so clear.
Continuity mistake: The Enterprise looks way more damaged than it did at the end of II. In the Wrath of Khan, Enterprise takes nearly all her damage on her port (left) side. At the start of The Search for Spock, she is damaged about equally on both sides, and still more so. (00:12:05 - 00:13:20)
Continuity mistake: When Kirk checks the video logs to find the keeper of Spock's katra, the timestamp reveals that Spock melded with McCoy on stardate 8128.78. The Wrath of Khan begins on stardate 8130.3. (00:21:50)
Continuity mistake: When David and Saavik approach Spock's tube on Genesis, you can see quite a bit of dirt covering the words' red lettering, but when it does another shot of the lettering, it is noticeably cleaner. (00:30:25)
Continuity mistake: Early in the film when a lifeform was detected in Mr Spock's quarters, you see a blip on the screen inside the old Enterprise, not the new Enterprise, which was almost a complete rebuild, with a very different layout. Notice the outline of the ship - different engines. (00:13:30)
Continuity mistake: In the scene early in the movie where Kirk is talking to Sarek the guns on the wall are suddenly rearranged. (00:23:50 - 00:24:45)
Suggested correction: The stardate system has never been precisely defined, so this is not a mistake.
While not precisely defined, it has been established in almost every episode that the numbers increase as time moves forward. (example in the Next Generation, the second number of the stardate corresponded with the season number, also in the episode The Best of Both Worlds, Picard gives a Stardate of 43992.6, then later 43996.2. So this mistake stands.