Right after we see Shane in the beauty supply store helping Rory and Lane, we find out that she was in the closet in Luke and Jess's apartment. Later that same day, she is still back at the beauty supply store, talking to Jess about what time she gets off work. She couldn't have been at work and in the closet at the apartment at the same time.
When Lane is getting her hair dyed, the shot switches back and forth as she and Rory talk. The purple dyed areas change back and forth, blonde, some dye, no dye over and over.
When Rory is applying the bleach to Lane's hair in one shot the bleach is covering just half of her hair, then in the next shot there is bleach on all of her hair, without Rory having applied it.
When Lane and Rory leave the beauty store, they are both wearing different clothes than when they're in Lane's house, even though it's supposed to be the same day. Shane is also wearing a different outfit when Rory and Lane buy the stuff to dye Lane's hair than when she's talking to Jess on the phone and Rory yells at her.
In "One Has Class and the Other Dye" Lane is angrily ranting and Dave drags her out to talk to her. Lane grabs her coat but it's gone by the time we see her outside.
When Lorelai is on the phone with Debbie agreeing to speak to the class, you can see behind her into the living room and a crew member walks into the shot.
When Lorelai and Rory start walking in the end, a boom mike can be seen from the window as a reflection.
When Luke and Lorelai walk into the classroom, a shadow coming from the boom mike can be seen on the wall above them.