Lucas's tattoo disappears and reappears after the car crash. Most of the time it is covered though because he is wearing his basketball uniform.
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.
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.
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).
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.
Throughout the series, it's clear that the actors/actresses portraying teenagers are actually in their 20s+. This is done to get around child labor laws, that won't allow teenagers to work full time on a set due to interfering with their real-life schooling. Plus casting adults adds to the benefit of better acting from the actors/actresses due to having more experience plus the producers are able to sexualize the characters without consequences. Lastly, it gives the general viewing audience who are teenagers (or young people) someone they can look up to.
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.
When Hayley and Nate flash back to getting married they're on a beach in the day time, but in the time line of the show they get married at night after the playoffs game.