When Lorelai and Rory are at the grandparents' house, Rory is seen picking up her drinking glass from the table in the middle of her and Emily. But when the shot changes to Emily, the drinking glass is still on the table untouched.
When Lorelai and Emily are listening to Paris talk at the swearing in ceremony, Emily makes the motion of checking her watch, whereas it is very obvious that there is, in fact, no watch on her wrist.
When talking to Jess at Doose's market, Rory says "I could care less about you and Shane". This is a grammatical mistake. The correct phrase would be "I couldn't care less...". A smart and intellectual student like Rory wouldn't make this kind of mistake.