Continuity mistake: In the final duel, when Gary Cooper is shot on his left arm, he grabs the spot where he was supposedly wounded with his right hand, then takes off the hand, and we don't see a wound and his shirt is not torn. In the following shot we see both the wound and the tear in the shirt.
Plot hole: Frank Miller sets fire to the stable Kane is hiding in; when he leaves it, you can see the large column of smoke. But the fire is ignored after that. In an age of wooden buildings and poor fire equipment, that could have burned down the whole town. Even assuming that people wouldn't come out to fight the fire during the gun battle, you'd think that when they all came out at the end, someone would have pointed out they had a major fire to deal with.
Continuity mistake: After the wedding Will Kane lifts up his new wife and seats her on a sideboard. Then, after getting the kiss he asked for, he swirls her around and puts her down on the floor again. In this shot we can only see their upper bodies, but from the way the downward movement pauses while their heads are still almost at level, we can tell that she must have stepped on a stool or something before she touches the ground.
Continuity mistake: The town people are gathered in the school/church with the doors open, then they dismiss the children and close the doors. The scene cuts back and forth between the Gary Cooper and the people sitting in the pews. The children reappear with the doors open and then again gone with the doors closed.
Continuity mistake: In the final scene, after the shootout, when Kane takes off his star and flips it to the ground, the flip starts at his waist and goes out at an angle, but when the camera descends to his feet, the star is only three or four inches from his shoe. Clearly, there was a star already on the ground and the star that was flipped went much further out and out of range of the camera.
Other mistake: There's a minor technical error in the portrayal of the telegraph equipment at the train station. When a message was coming in for the stationmaster the repeater relay, on the left, was shown clicking away. The telegraph sounder to the right wasn't moving. Both should have been in use as the repeater relay would have been fairly quiet while the sounder would have created the loud audible clicks. Repeater relays with fresh sets of batteries permitted the telegraph signal to travel greater distances by regenerating the Morse code signal at each telegraph office. The sounder made it much more audible. An office at the very end of a telegraph line would only have a sounder.