Revealing mistake: When the Federation fleet is preparing to warp to Vulcan, you'll see the first ship blast off, but the second ship blasts off out of nowhere. You could argue that the ship was out of sight and below the Enterprise, but the light trail isn't that long. The light trail starts in front of the Enterprise where you should be able to see a star-ship. (00:40:45)
Revealing mistake: At the end of the film, when Kirk is receiving his medal, in the background you can see the female alien crewmember from the Enterprise bridge clapping with very human-like hands. (01:56:15)
Answer: Kirk's never met Uhura - he wouldn't know where she's from. When he asks her name, she says that her name is "just Uhura" - Kirk's expecting to hear two names, first name and surname (just as he introduced himself as "Jim Kirk"). As such, his first question is to ask whether they don't have surnames on whatever world she comes from. As it happens, she is from Earth, she just doesn't want to tell him her full name; he doesn't know that, so he's making assumptions that are, in this case, completely wrong.
Tailkinker ★