Deliberate mistake: Robert Brown wears a gold turtleneck sweater with his leather outfit throughout the series. Though it's more than 100 years beyond the show's 19th Century setting, he and the producers decided that he looked good in it, so allowed the anachronism to stand.
Deliberate mistake: When Mary gets to the school where she is going to be teaching, all the children are standing outside wearing shoes. However when they come in to the school they are barefoot. I have a feeling this was done to show how poor the community she was teaching in was supposed to be, but it looked like too cold and rainy of a day to let the actors be outside barefoot.
Deliberate mistake: At the end, the U.S. Cavalry "brand" Blue finds on the mule isn't a brand at all: it's obviously been applied with white paint to make it more visible to the camera. It's also unlikely that it would be so large, or that it would be located on the animal's neck instead of on its rump. (00:46:50)
The Night of the Tottering Tontine - S2-E16
Deliberate mistake: The knife that has just killed one of the investors doesn't have a trace of blood on it. No one had time to clean it off. (00:20:00)