Continuity mistake: The professor mentions he can't drink vodka with the Russians because he's allergic to alcohol, yet in the episode "Man with a Net" (with Lord Beesley), he drinks a strong alchoholic drink he made along with everyone else with no signs of any sickness, or reaction preventing him from drinking it.
Continuity mistake: We see a shot of the outside of Ginger's hut when she and the Skipper are first planning for the beauty contest. The door to the hut has a window in it and we can see the Skipper's hand leaning on the sill. We cut inside and suddenly Ginger is by the door and the window has vanished.
Continuity mistake: After the shark bites off the rest of the paddle, leaving the handle, Gilligan hesitates a second, sees no paddle, and begins swatting at the shark. A moment later, in the closeup, he's paddling, hears the bite again, and pulls the handle up, showing the paddle is gone.