Here are a few possibilities
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Captain James T. Kirk. This is the casting I read and really liked. Replacing the legendary William Shatner, Matt Damon. The guy can act, he can do action, he can shift between comedy and drama with ease, and I think he could do the character proud. The big question will be if he would go the Shatner route, or if he would employ more human-like inflections. Either way, I think Damon would be a good choice.
Commander Spock. Originally played in stoic, emotionless fashion by Leonard Nimoy, he always had this great matter of fact way of delivering lines, and having great presence with minimal motion. Who could don the pointed ears and take over as the half alien science officer? I read someone suggesting Adrien Brody, I don't think so. The name that I like is Billy Zane, he has the right voice, and facial features that look right for the role. He doesn't really look like Nimoy, but I think I could believe him as Spock.
Doctor Leonard "Bones" McCoy. There is only one person I can think of to take over the reigns from the late DeForrest Kelley, that name is Gary Sinise. He can act and has the perfect look to take on the role of the good doctor. Can't you picture Sinise entering into some verbal sparring with Zane's Spock, or one of those "I'm a Doctor" lines, or even a classic "He's dead, Jim." Sinise is the one name to consider.
Chief Engineer Montgomery Scott. None other than Austin Powers himself, Mike Myers, would be up to the task of the late James Doohan's most famous role. Take this comedic chameleon and focus his abilities on the Scottish engineer, and you could have a match made in heaven. There is no doubt that he could do the accent, and he seems to have the right physical presence for the job. Myers in this role could prove to be inspired casting.
Ensign Pavel Chekov. This is a tough one, a young Russian for the helm. Perhaps someone like Frankie Muniz? He seems to be about the right age and have the right build for it. I think the biggest question would be if he could pull off the Russian accent. I am sure a vocal coach could get him doing it, but then it would be up to the audience buying Malcolm as a bridge officer on the Enterprise. Could be interesting to see if he can put a spin on Walter Koenig's Monkee inspired role.
Lieutenant Uhura. Another tough one. How about Regina King? She is a very good actress, and I think she could take on the role of the Enterprise communication officer. She is equally adept at drama and comedy, as shown in films like Ray and Miss Congeniality 2. I would like to see her in the role originated by Nichelle Nichols.
Lieutenant Commander Hikaru Sulu. As I was thinking of names, a certain guy came to mind, but I could only think of the movie I saw him in, not the name. The film is Better Luck Tomorrow and the actor is Parry Shen. I know, Shen is Chinese and George Takei is Japanese, but that doesn't chnage my opinion that Shen would be very good in the role. I think he has the right look, and from what I saw in that aforementioned film was very good.
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Don't like the Cekov choice, not quite sure about Myers as Scotty either. But I do like Billy Zane for Spock alot and Gary Sinise could work great as Bones. As for the other they seem pretty good choices.