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.
Magical Express Murder Case: File 3 - S2-E12
Character mistake: In Akechi's flashback, the board with the flight for Los Angeles has accurate names of airline companies (they seem to have swapped "Virgin" with "Kristen", for some reason) but the flights for Seoul is from "KSREAN" Air, clear misspelling of "Korean." (00:14:15)
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.
Convictions - S7-E2
Character mistake: In the previous episode, Bernice was arrested because the police thought she was Jimmy's client, and Eugene wouldn't allow him to refute that, since it would reveal the true kidnapper, their client, is still out there. In this episode, with Jimmy now representing Bernice, he admits in court she wasn't his client, but despite Eugene's concerns the cops don't put two and two together and continue to pressure Jimmy to reveal the real kidnapper, who resolves the story herself by coming forward.
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.