In the early episodes, Luanne clearly has automotive skills - she is seen repairing cars and even ordering Hank parts he needs to fix his truck at one point. From "Jumpin' Crack Bass" on she has little to no knowledge of cars.
In earlier episodes, it's stated that Peggy grew up in Arlen and her mother was basically an older version of her. Later episodes change her backstory to say that she grew up in Montana, with a mother who was a harsh rancher.
The picture of Cotton Hill in Hank and Peggy's bedroom changes from episode to episode. In some episodes it doesn't even resemble Cotton.
Bill's house is often shown in front of the Hills' house and is often shown in the back of the Hills' house as well
Hank's senior year of high school was 1974 according to some episodes and 1979 according to others.
In many of the shots of Game Boys and video game controllers, the buttons are on the wrong sides.
Through out the early seasons when Hank has the Ford Ranger, the transmission changes from a manual transmission to automatic several times. As well in some episodes when its portrayed as an automatic, the gear shifter is on the floor. No Ford Ranger stock has the gear selector been on the floor, only the steering column.
In certain episodes, the Hill family goes to the hospital called Heimlich County Medical Center. In other episodes they go to the same hospital (according to looks) and the sign outside reads Arlen McMaynerbury Medical Center, not Heimlich County Medical Center. However, it is not uncommon for public buildings (hospitals, schools, convention centers, recreation centers, etc.) to share common designs. There are likely many medical centers of this same design all around Arlen, McMaynerbury, Durndle, and the other cities in Heimlich County.
When Hank was a senior in high school in (what according to his calendar is) February 1975, he had a Jaws movie poster on his bedroom wall. Jaws wasn't released until the following summer.
Hank's drivers license lists his address in Arlen TX with a zip code of 73104. Although Arlen TX is a fictitious town, the zip code is real. However, the zip code they chose is in the wrong state - 73104 is in Oklahoma City OK.