When the drill sergeant was announcing who was accepted into the Navy, his hands are yellow for a few frames, while he is usually dark-skinned.
When Homer and Lisa are sitting in the living room, Lisa is seen wearing her nightgown. A few seconds later it changes to her normal dress without an explanation.
In episode 3G04 "Simpson Tide", there are numerous Navy mistakes. - Women are not allowed to serve on a submarine in the US Navy. - Junior enlisted do not wear khaki uniforms. - Homer's drill instructor seems to be a marine corps gunnery sergeant and not a Navy Recruit division commander, which wear red ropes in uniform. - Capt. Tenille was wearing the uniform of a lieutenant. - Rating badges are not black on white uniforms. - Khaki uniforms do not have rating badges. - Khaki uniforms, as the female officer was wearing, do not have shoulder boards. - The collars on the dress white uniform are wrong.
Homer is given a "Dishonorable Discharge" after all members the entire court martial recuse themselves for various personal reasons. A Dishonorable Discharge is basically the equivalent of a conviction in a non-military court. Since the court martial adjourned without delivering a verdict (A punitive discharge must be handed down by a court martial), the worst discharge he could get would be an administrative one, likely an "Other than Honorable" discharge.
During Homer's court martial, the American flag displayed has seven white stripes and six red stripes instead of the seven red stripes and six white stripes. Also, when an American flag is hung vertically, the union (stars) should be at the top to the observer's left, not on the right as displayed.