As far as plot holes go, I think the plot holes in any movie will become apparent when it's viewed multiple times. Even Casablanca is full of continuity errors and has a few plot holes that become very easy to spot after the tenth viewing or so. Hardly ruins the movie for me, though.
You know what the difference between Casablanca and Star Trek is? Casablanca is a wonderful movie that wasn't afraid to take it's time with the story, it developed it's characters through many multiple methods and gave our cold hearted character a sense of morality.
Kirk started out as an a**hole, and stayed a**hole. And which was the better romance? The story story rich Rick/Ilsa romance, or the out of the blue Spock/Uhura romance? At least when Ilsa was in trouble, Rick did something about it. If Spock gets berated and insulted, Uhura just stands there and watches.
Star Trek is Plan 9 in comparison to Casablanca.