II (The Samurai called Jack) - S1-E2
Revealing mistake: When Jack cuts the flying car's hood off, he slashes it vertically, but it is cut horizontally.
XXXVII (The Birth of Evil Pt. 1) - S3-E1
Continuity mistake: Jack's father the emperor is a young man when he defeats Aku. At the conclusion of the movie, he greets his pregnant young wife. At the start of "The Beginning" movie, Jack is a very young boy, both his parents are well past middle age-and Jack has no sisters or brothers to explain the gap.
XX (Jack and the Monks) - S1-E20
Revealing mistake: Immediately after the time portal is destroyed, it stops raining but there is no moisture anywhere.
Continuity mistake: When Jack meets the old father, watch the platform thing he is on. The yellow/black pattern alternates on the right side.
II (The Samurai called Jack) - S1-E2
Continuity mistake: Right before the three-eyed waitress leaves to talk to Aku, there is a shot of an octopus-like bartender, whose facial expression changes in each shot.
Factual error: Even though the show is called "Samurai Jack", Jack isn't really a samurai. The correct title for him would be a Ronin as a Ronin loses their master if their master dies or loses favor with their master and also has no home anywhere. Jack has these attributes. He has no master and no home.
Suggested correction: Nor is his name really Jack. "Jack", and thus "Samurai Jack", is just an alias that the prince takes. It's not meant to be his title or rank. Just like Cowboy Jack wasn't actually a cowboy.