Plot hole: In the Upper Class Twit of the Year Show, there are five contestants. However, after Oliver runs himself over, in the events that follow there are only four props for the remaining twits (four mannequins, rabbits, and guns); since Oliver's death was unforeseen, shouldn't there be five of each? (This mistake, by the way, is rectified in the film version of this sketch).
Career Fever - S4-E9
Plot hole: Peter thinks he has a fatal disease. The parents realise two pages stuck together and he went from page 95 to 98 mistakenly. However, Mike first read about the actual fatal disease info on page 97, and then Carol read about what Peter actually had on page 96 (poison ivy). If Peter read only 95 and 98, there would be no way of Peter to know his disease was fatal since that was printed on page 97.
Plot hole: In Blakey's office Stan shows Blakey the toy bus he has just brought for his nephew. How can Stan have a nephew when he isn't married and his only sibling is his sister Olive, who doesn't have any children?
My Late Lamented Friend and Partner - S1-E1
Plot hole: When Jeff and Marty are parked, waiting for Sorrenson to exit the building opposite, Jeff says he should call the police. But he immediately follows Sorrenson and has no time to do so. Later, they arrive even though not called.