Deliberate mistake: About halfway through the show, Roz and Daphne are in the kitchen talking. Daphne pours water into the teapot - but there is no steam coming from the kettle, nor any sound of a boiling/just boiled kettle. No self-respecting Brit would make tea with water that wasn't boiling.
Janitor Dad - S4-E6
Deliberate mistake: Chet comes home from his first day as school janitor brandishing a paycheck, which impresses his wife and in turn changes Shawn's mind about Chet working at the school. Being that it was Chet's first day on the job, there's no way he would already have received a paycheck. Even if it was payday, his employment would have begun on the first day of the next pay period.
Suggested correction: It's possible it was pay for training or orientation days that were before his official start date.