Character mistake: Winona says that Jason killed a penguin on their trip to the Arctic. This is impossible because penguins only live in the southern hemisphere.
Character mistake: The math test Tina gets a D on has #5 marked wrong. However it is correct. (Additionally it seems improbable she would actually get #3 and #4 wrong since she obviously understood the concept of how to do the problem and only had a problem with the division for some reason, and these questions didn't require division).
G.I. Jessie - S2-E26
Character mistake: In the beginning of the pilot episode, Jessie told the cab driver that her father is in the Marine Corps. However, it is shown in this episode that her father is actually a lieutenant colonel in the Army. It's highly unlikely that Jessie wouldn't know what branch of the military her father was in considering how much she spoke about her military upbringing throughout the series.
Character mistake: Iain and Mickey tell paramedics about an old lady needing immediate help in her apartment on the "5th floor" just before stealing their ambulance as the paramedics storm into the building. However, the apartment building has lots of units on every floor and no unit # was given in this fairly large building. Medics would have trouble finding her had this been a real call.
Character mistake: At the start, when Adam messes up Johnny's hair, it cuts from him outside the dining room door to him at the dining room door with no time to sort his hair out but his hair suddenly becomes perfectly done. (00:01:36)
Character mistake: Max enters Brad and Jane's house to find Brad dancing with Santa, Max then calls Brad "Dave." (00:09:18)
Character mistake: Mike is holding and talking about his .59 caliber flintlock and refers to placing a percussion cap on it to fire it. A flintlock does not use a percussion cap.