The Curious Tale of Mr Spearfish - S3-E1
Character mistake: The whole mystery kicks off when Creek sees a photograph of Alice Spearfish with her parents, and concludes that she is adopted because two people with brown eyes cannot have a child with blue eyes. An obsessive polymath like Creek must surely know that that is wrong. The genetics are complex and though uncommon it is possible for a couple with brown eyes to have a child with blue eyes and the other way around.
Character mistake: When the zombies break into Buffy's house, she stakes one to check if it's a vampire. But she should know full well that it isn't a vampire as they all broke into her house without being invited.
Touchstone - S2-E14
Character mistake: The team are finding out who could have stolen the weather device from Madrona and find an energy spike at the same time SG2 are on a mission. "But SG2's mission is galaxies away from Madrona" Says Teal'c. At this time the only missions the teams could go on was in our own galaxy and couldn't be galaxies away.
Simpsons Already Did It - S6-E7
Character mistake: Butters says that he watched "all 132 episodes" of the Simpsons; at the time this episode aired, 291 episodes of the Simpsons had been produced and aired.
Suggested correction: Butters is always shown as being a bit dim and slow. As a kid it's easy for him to make a mistake in thinking that there was only 132 episodes.
That's why it's called a "character mistake"; also, Butters is certainly naive, but he's hardly what I would call "dim and slow."
Character mistake: When Mikey looks at the Picture Day schedule, he points out that 4th grade is going last. However, the schedule says they are after 3rd grade and before 5th grade, actually making them fifth. The faculty is actually listed last on the schedule.
Character mistake: Ben Stein asks a chemistry question about -273° Kelvin. The problem is, the scientific community never uses the word 'degrees' when referring to Kelvin. The correct way to say it would have been, "-273 Kelvin is also known as what?" Also Kelvin cannot be negative since 0 kelvin (approx -273° Celsius) is absolute zero.
Wax Doll Castle Murder Case: File 2 - S1-E8
Character mistake: Hajime's proposed solution is obviously wrong, but not for the reason they mention; the 'killer' could never clean themselves up completely and effortlessly in such a short time from the soot.
Character mistake: When Walker and Satch are in the car looking for Walker's son, Walker tells Satch his son is 17. Afterward, when they are out on the streets, Walker asks some people if they have seen his 18-year old son.
Suggested correction: Jonathan actually states it's a photo of 2 blue eyed parents and a brown eyed daughter.
Which is still possible! It isn't even uncommon. Brown eyed parents can have a blue eyed child and vice versa - that's what recessive genes are all about.