Although the 37-star flag suggests the setting date is within the 1867-76 frame, the date on the large boarding house at the end of the street as Amy and Helen are taking the buckboard out of town is "1888".
The pre-gummed rectangular envelope that Kane puts his folded Last Will and Testament into was not the type used in the 1867-76 era that the 37-starred US flag suggests and was not made until about the 1890s. Envelopes in that era were diamond-shaped with the user having to provide the sealant (wax, for example).
In a number of scenes there are Pabst Brewing signs seen on the inside and outside walls of the saloon. Although Pabst did brew in 1848, it did so under the name Best and Company and did not change to Pabst until 1889; the 37-star flag suggests the setting dates are between 1867-76.
Just after Kane leaves the barber shop and walks up the street, sunlight can briefly be seen reflecting off cars passing by the set in the distant background. It is reportedly more noticeable in the theatrical release than on TV screens.
In the church scene, the front wall behind the pulpit is illuminated with wall-mounted candelabras, each one holding what appears to be lighted candles. The church scene is an extended scene, but throughout the scene, none of the candles' flames flicker, waver or smoke, as would be typical of real candles. These are obviously electric candles with electrically-powered bulbs, but judging by the setting and time period, it's clear that the story takes place before modern electricity.
When Kane scribbles the note saying "Back in 5 minutes", the frontal camera shows the sentence beginning with "Back" being written at the top left of the page. However, the view changes to a camera behind Kane and he is still writing, but this time we can see the "Back" has begun halfway down and nearer the center of the page.
When Kane and the judge are arguing while the judge packs, as Kane asks, "Why?" in close-up his head is turned slightly to his left. It then cuts to a longer shot and his head is turned visibly further to the left.
When Will and Amy start leaving in the beginning, they appear to travel a substantial distance away from the town, but when they get back, the clock shows only ten minutes have passed. This would be virtually impossible for a horse-and-buggy ride.
When Kane and his wife are having their argument in his office, he is checking his gun to make sure its fully loaded. During the few seconds he's doing this, the gun is pointed directly at his wife's chest. Maybe not a goof, but certainly not a very wise thing to do, to point a gun at another person, while the gun is fully loaded.
At the final stage of the shootout, just after Amy / Grace Kelly scratches Miller's face, Kane / Gary Cooper shoots Miller twice. Not once the gun barrel is actually aiming at Miler at the moment of the gunshot: it is always aiming to the ground some feet in front of where Miller / Ian McDonald is standing.
The goof items below may give away important plot points.
In the climactic gunfight, after Kane has shot Ben Miller, we see Kane running between buildings into a back alley area off the main street of town. He stops by a tree and looks back to see if he is being pursued. As he sets off again, we see the back of a brick building with an air conditioning unit mounted on the outside of a second-story window.
When the barn is set on fire, it can be seen burning with lots of flames and smoke. Less than five minutes later, and during the final scene, Kane and Amy ride right by the barn in the background and there is no damage and no smoke.
After Amy shoots one of the bad guys in the back, there is a shot of Will Kane looking out a window holding his gun in his left hand. There is an immediate cut to a shot of him holding the gun in his right hand. The left-handed shot appears to have been done to make the composition of the shot more dramatic.
When Kane approaches the church to ask for help from the people inside, a very large column of black smoke can be seen in the background just to the right of the church. That smoke was from the burning barn that is seen later on in the film, although it was actually shot before the church scene.
As a result of the barn fight scene between Kane and Pell, Kane is left with a large abrasion on the right side of his face and goes to the barber shop to get cleaned up. The abrasion is gone while he is in the barber shop, but it can be seen in subsequent scenes.
Once the showdown began, the first time Kane is fired upon by an off-screen gunman, the bullet strikes the side of a barn about a foot over his left shoulder. At the same time Kane grabs his upper left arm as if he is wounded. His shirt from that point on is torn as if damaged by a bullet. The ballistics involved for that scenario just don't add up; the bullet would have had to bounce off Kane's arm in an impossible trajectory.