Deliberate mistake: Marian is wearing pantaloons, or knickers, in the rape scene. Knickers were not worn by women until Victorian times, and then they were split at the crotch.
Deliberate mistake: It's not particularly believable that the away team to stop the assassination would consist of the ship's entire senior staff, and raises the question of who was left in charge of the bridge (the answer: apparently nobody, as they return to a completely deserted bridge). Of course this is dramatic license, and at this point the audience won't care or likely even notice.
Deliberate mistake: When Ed's looking at the drawings done by the sketch artist near the start, he's flipping over a book of them. Trouble is that they all end up facing the camera, meaning that Ed must have been looking at the blank side - we see the blank pages briefly as he flips them. (00:14:30)
Deliberate mistake: Smoke from a smoke machine smells very acrid, is standard temperature and would never be confused with hot steam from a shower. The man with the knife would have suspected something is wrong once he was in range to smell and feel the "steam". (00:26:35)
Deliberate mistake: When Cliff crashes into the pond after his first flight, Peevy tells him that he has to look around with his eyes and steer by moving his head. However, just before this as he is flying next to the passenger airplane, he looks over at it, turning his head to the left without changing directions at all.
Deliberate mistake: When everyone is surfing at night, the sound of the waves in the background is significant, as it should be. However, at one point, when Keanu and Lori are having an intimate conversation on the water, the sound of the waves disappear. The wave sounds resume in the next scene.
Deliberate mistake: While in the Japanese bath you see the fat samurai swim in the "pool" with an underwater camera. But when on the surface you can see the "pool" is no more than a man's length, in other words he could not have been swimming in there. (00:42:45)
Deliberate mistake: When Lithgow is shot in the leg and trips backwards over the crate or box, the stunt man is deliberately visible. It's not Lithgow making the fall.