Edward Mordrake: Part 1 - S4-E3
Deliberate mistake: After Elsa kicks Bette and Dot off the stage, Elsa looks at the sheet of music on the stand and you see the name of the song 'Criminal' and it also states it's by Fiona Apple which is true, but this season is set in 1952, and Criminal wasn't released until 1996.
Episode #3.6 - S3-E6
Deliberate mistake: When Fidel pops behind Humphrey with the walking cane, in nonsensical fashion his cane switches hands in the brief moment when the camera moves from behind him to the front of Humph. It has to be deliberate since there is no cut, but Fidel is standing still both times, and Gary Carr must have done a sudden swap that has no reason. (00:28:20)