Audio problem: When John Wayne has his back turned and tells the newspaper man "print that and he will kill you sure as hell", you can tell by the side of John Wayne's face that he isn't really saying anything.
Character mistake: Stoddard in class says the founding document of the country can be added to with amendments. Pompey says it is the constitution written by Thomas Jefferson. Stoddard says 'declaration of independence'. Pompey was right, Stoddard incorrectly corrected him. It IS the constitution. The declaration can't be added to with amendments.
Continuity mistake: The colour of John Wayne's shirt changes from black to grey after entering the burnt out building. It can not be dust, as the covering is far too even and complete, and a couple of shots later it is black once more.
Continuity mistake: At the very beginning of the movie, we see a train rolling down the hill, the first wagon has a dark door in the middle. In the next shot, we see the train coming in the station and the first wagon does not have a door on the middle part of it, but at the beginning of the wagon. Obviously another shot, another train.
Continuity mistake: During the election meeting held at the saloon, there's a large sign that reads "Capitol City Here We Come" on the wall by the stairs. We see this sign in numerous shots. When Valance enters the saloon there's still a sign on that wall by the stairs, but now the painted handwriting on this sign is different. Moreover, the original sign that had been by the stairs is now up on the side wall of the saloon, where Valance and his thugs stand; note the handwriting. (01:09:20 - 01:13:45)
Revealing mistake: When Ransom and Hallie Stoddard are riding the train back to Washington at the end of the film, pay attention to the view out the window as Hallie responds to Ransom's suggestion of returning to Shinbone. When the conductor brings the spittoon and talks to them, the same exterior footage repeats from when Hallie started to speak. (02:01:30 - 02:02:15)
Continuity mistake: Toward the end of the movie in the meeting hall, all the good guys are seated on the left and the bad guys are on the right, when looking from the stage. After John Wayne has his talk with James Stewart outside the meeting hall the doors kick open for Stewart to go back to the meeting. The good guys and bad guys in the audience are now on opposite sides of the meeting hall as you look toward the stage from the rear.
Other mistake: While Ransom Stoddard and his wife are on the train back to Washington, the conductor tells them that the train will be able to maintain a speed of 25 mph throughout the journey. The film is set around 1900, by which time trains were regularly topping 65 mph, and the outside scenery is moving faster than it would for something traveling at 25 mph.
Answer: At first glance, it looks like the word "LEFT" in large, widely-spaced lettering. It's very unlikely that this would be a blocking cue, telling Jimmy Stewart to move left (which he does from that point). That's not the way blocking cues are done, for one thing, and a seasoned actor such as Jimmy Stewart wouldn't need such a cue. It's also very unlikely that it's a warning message from John Wayne who is hiding in an alley across the street, on Stewart's right. John Wayne didn't want anyone to know that he actually killed Valance, so he wouldn't alert Jimmy Stewart with any messages scrawled in the street. All hypothesis aside, it's probably just footprints in the dirt, an illusion of light and shadow.