Revealing mistake: When Chekhov falls on the aircraft carrier a cloud of dust flies up as if he's landed on a mat or soft cargo. When we see him on the floor it's concrete.
Revealing mistake: When the whales first swim past the now submerged Klingon Bird of Prey, if you look to the right where the water is supposed to meet the distant horizon, you can see an exposed edge to the 'horizon' as the water moves up and down. This is the edge of the pool they are in, and the exposed edge is the bottom of the painting of the sky behind them. (01:46:30)