Near the end of "The B- Team of Life", When Amy is walking down the stairs saying that Morgan wants to hear a Story from Corey, There is a boom mic on the top of the screen right in front of the shelf to the right of the stairs.
At one point, Cory states that Mr. Feeney has been his teacher since first grade. However, in the first episode Eric tells Cory that he's going to have Mr. Feeney now that he's in sixth grade, suggesting that he did not have him in previous grades.
In the second season (Cory's first year of middle school), the locker right next to Mr. Turner's classroom changed ownership throughout the year. It was Frankie's, then Cory's, then Shawn's, and then Eric's. Every time it is shown, there are kids in it but it belongs to a different person.
In several episodes, the names and identities of Topanga's parents change. In season one, she calls them Jedediah and Chloe. In later seasons, they are Jedediah and Rhiannon. In total, four men and three women have played Topanga's parents, including a casting change right in the middle of a story arc regarding their divorce.
The main characters were in the sixth grade and eleven years old when the series began in Fall of 1993. They should have graduated from high school, therefore, in the year 2000. Instead, two years were missing from their high school careers and the writers had them graduate in 1998. When season 2 began, they were in 7th grade and this was referenced by head bully and repeating senior Harley Kiner. He was known for especially picking on seventh graders and kept reminding Cory of that fact throughout the second season. Somewhere between season's 3 and 4 their year in school became hazy between them being in 8th grade and freshmen in high school.
In the episode "Hair Today Goon Tomorrow" the guy who gives Topanga his card, his hair changes between the shots, in one shot there's a piece of hair in his face and the next it's gone, and then it's back again.
As with any long running series, certain details change as the show progresses and writers change. Both Shawn and Topanga, for instance, have sisters in the first season but are later depicted as only children. In the show's earlier years, Topanga's mom was named Chloe, but in later episodes she's known as Rhiannon. Inconsistencies like these are generally forgiven in series.