Set in 1912, the song "Trouble" mentions both the beverage Bevo (first offered in 1916) and the magazine "Captain Billy's Whiz-Bang" (first published in 1919). "Whiz-Bang" is named for a type of artillery round in World War I (1914-1918)--the publisher was a veteran of that war.
When Harold Hill tells Mrs. Paroo about the "great cornet players ... O'Clarke, O'Mendez, and O'Klein", he's referencing the (obviously non-Irish) real life trumpet/cornet virtuosos Herbert Clarke, Rafael Mendez and Mannie Klein. Mendez and Klein would have been around six and four years old at the time of the story.
When Winthrop asks Harold Hill if his proposed uniform has "a stripe", Harold assures him that it will. But when the boys come into the schoolroom wearing the newly-delivered uniforms, the uniforms very clearly do not have a stripe on the pants. However, the uniforms worn by the marching band at the end do have a stripe.
Right before Shipoopi, Mrs. Shinn says her girls are ready to perform, but Ethel dances with Marcellus in the number and is dressed in a flowered dress. Immediately after the number, she is dressed in her Grecian urn outfit, ready to perform.
In the candy shop when Hill and Marion are sipping on their strawberry sodas, Hill releases his straw and the straw floats to the top. The camera angle changes with Hill holding the straw and it's at the bottom of the glass.
Chairs in front of the stage move around during the July 4th celebration in the auditorium. First there are three together with one chair a ways off to the left; then later, as one of the school board members prepares to sit down, the four chairs are in a group.
Robert Preston's hair goes through several color changes through the course of the movie. At the beginning it's a sandy blond, then it grows darker and by the time of the footbridge scene it's a deep, chocolate brown. Yet minutes later when Prof. Hill is apprehended in front of the Paroo residence it's back to sandy blond.
After waving to the departing salesmen on the (yellow) train he came in on, Harold turns to leave the depot and the only thing behind him is a lonely (brown) freight car sitting still on the track. This same freight car is visible there many days later when the Wells Fargo Wagon comes into town, and four weeks later when anvil salesman Charlie Cowell is leaving town.
In the opening sequence, the train is traveling from the Illinois Iowa. The background visual scroll only shows flat cornfields. We never see it crossing a river, let alone the Mississippi, which is the only way to get between the states in question.