In the first season Troy says he injured his shoulder in a keg flip. But it is changed in a later season to his knee.
The Colorado Bar Association automatically disbars anyone who lies about having a bachelor's degree for a minimum of eight years, not the conditional four-year probationary period Jeff is given to obtain said degree. Further, reinstatement to the bar after that period would be highly unlikely given his previous dishonesty.
Jeff is said to be attending Greendale for four years in order to attain his Bachelor's Degree. While the majority of Community Colleges only offer 2 year AA degrees, currently 17 states (including Colorado) allow Community Colleges to offer 4 year Bachelor's Degrees.
In early episodes, including the pilot, characters refer to the study room as "upstairs." However, in later episodes, it is made clear that the study room is on the ground floor.
Despite the show being set in Colorado, palm trees are plainly visible in some episodes.
Throughout the series, it is established that Shirley is a "Baptist" (general, but enough info); however, in various episodes she does the Catholic "Sign of the Cross" (signum crucis). If she were a baptist (any variation) this would not be a ritual of her faith. However, there isn't any prohibition about making the Sign, either, if she happened to feel like doing it.