Continuity mistake: When Laura pushes Mrs. Pierce, she lands in a pail of water on her buttocks, and her hair remains dry. The next shot is of Laura saying something, then in the very next shot we see Mrs. Pierce still sitting in the bucket, only now her hair is soaked. Since she landed on her behind, her hair should not have been soaked, and there was no time for anyone to pour water on her head between the shots.
Factual error: Laura Ingalls Wilder was born in 1867. Ralph Waldo Emerson died in 1882, which means she was at the maximum 15-16 years old (and already married) in order to meet him IF he were to appear in 1882 which is not possible because: The problems with his memory had become embarrassing to Emerson, and he ceased his public appearances by 1879 (according to wikipedia).
Continuity mistake: The restaurant has "Caroline's" painted on the windows (except while operating as Mrs. Sullivan's). However, near the start of the episode, when the horses trot by in a nighttime exterior shot of the restaurant, the window still reads "Nellie's" (taken from earlier in the series). This same mistake has been seen in at least one other episode of season 8 as well.