Lorelai's hairstyle continuously changes from curly to straight from scene to scene. First she's outside walking with Sookie and it's straight, then when she's inside at her house it's curly, walking to the town square it's straight, but then once she reaches the town square it's curly, at Luke's its straight again, but when she and Luke meet up outside it's curly.
When Luke leaves the diner to 'run an errand,' the door to the diner rings a bell as it opens. However, the bell on the door-frame is visible, and doesn't actually move.
When Emily announces their touring schedule in Rome to Rory, she says they'll visit the Piazza Navano during their passeggiata. There is no Piazza Navano in Rome: It's the Piazza Navona, something a sophisticated veteran traveler like Emily Gilmore should have known.