Last Train to the Fair - S1-E30
Factual error: At the end of the episode as the train heads towards Sacramento, above in the sky are jet vapor trails streaking across the sky.
Continuity mistake: When Joshua was breaking the horse his hat fell to the ground, change of camera to those watching and then back and his hat is back on. The shot change to watchers again, then back, and his hat is back on the ground. This happens a few times.
Factual error: A scene near the beginning of the episode shows a man talking to a crowd holding a Peace dollar. Peace dollars were not in circulation until 1921, and I believe that the setting for this series is around 1880.