Early on, the Sheriff states that Quake City was built on a fault line. Professor Lawson who first named the San Andreas Fault in 1895, did not identify it as running all the way through Southern California and for this to be the root cause of large-scale land movement causing quakes until 1906. Likewise, it seems stunningly unlikely that a local sheriff would have had leading-edge, technical knowledge of geological plate tectonics that would have been worthy of a PhD in this era.
When Amos and Theodore are handling the dynamite, they keep saying it will go off if it hits the floor. This is a flawed way of thinking. Dynamite requires a special chemical reaction to be set off, hitting the floor is not enough. However in a previous scene, the bank owner proclaims that the dynamite is "sweating" meaning that a simple drop could set it off.
Near the end, when Donovan and Dusty both land into the river embraced to each other, their hats topple off, and sail away downstream, where Donovan and Dusty would not be able to catch them in time. In the next scene (where they are kissing while standing in the water), their hats are placed back on their heads again just the way they were, as if they had never fallen off.
During the opening credits, Magnolia's seat position on the stagecoach keeps changing. During long shots, she is seated on the right side of the stagecoach; during close-ups she is seated on the left side of the coach.