Jamie was born on the day of graduation in the fourth season
and then in the fifth season they celebrate his birthday. The problem is that school is obviously still in session since there are eleven more basketball games left in the season. So they celebrated a few months early.
In the first episode of season 3, it is established that the lockers are in alphabetical order, explaining why Haley is between Nathan and Lucas. In several episodes following, Brooke's locker is shown to be only one locker away from the Scotts' lockers, which would be near impossible with her last name being Davis.
The zip code of Tree Hill is not consistent throughout the series. In season 3, it is shown to be 01982 (seen when Brooke addresses her letters to Lucas) and in season 6 it is shown to be 01928 (seen on the envelope to Sam from her birth mom and on the postcard to Brooke from Sam).
In the episode "Life As a Glass House," Haley and Nathan are studying. Haley gets up to leave, she puts her bag on across her chest. A few seconds later she is seen, putting her bag on again except she keeps it on her right shoulder. And when she leaves, the bag is once again across her chest.
Nathan is seen wearing a blue shirt under his white uniform. A teammate is wearing a white undershirt. In the rules of national high school basketball, it is illegal for teammates to have different colored undershirts as well as having a color that is not the same as their uniform.
Peyton's full name (as seen mentioned many times) is Peyton Elizabeth Sawyer. Peyton says to Haley at one point that Brooke has been calling her "Peyton Marie Sawyer" because of the way she was acting. Peyton then explains that the initials spell PMS revealing that it's just a joke between her and Brooke. Therefore, Peyton's middle name does not change at any time in the series.