The Illusion of the Deadly Conglomerate - S1-E18
Continuity mistake: Blake tosses the trick card, and it lands on the lower right side of the target. When Dominick walks over to pull the card free, it has moved itself to the upper left portion of the target. (00:29:20)
Beyond the Farthest Star - S1-E1
Continuity mistake: There is an alien in a Starfleet gold shirt sitting next to Sulu. For a second, there is a human in a Starfleet red shirt, then it goes back to the gold shirt alien.