Rory, when given ID at the newspaper, states that she's never had photo ID before. On her first day at Yale she was given a photo ID, and would also have a photo on her driver's license
When Logan takes Emily's sewing box, he places it in his right jacket pocket. When Lorelai makes him give it to her, he takes it out of his left jacket pocket.
Emily gets Lorelai's drink wrong by asking if Lorelai takes her martini with vodka and a twist and Lorelai corrects her saying she always has a gin martini with an olive. However, Lorelai once ordered a vodka martini at a bar with Emily and has requested martinis with a twist at Friday night dinners. Once Emily even asks if Lorelai wants an olive and Lorelai requests a twist instead. For a long time Lorelai also asked for white wine at these dinners rather than a martini.
When Rory talks to Lorelai about her newspaper internship, Lorelai walks past the same place, even passing the same guy, twice.
When Logan and Rory are talking about how Logan's lack of knowledge about his own father, he says they've had two conversations, one being "Logan, when you're sailing close hauled, wait until you gain that last bit of boat speed before you pull in the Jib sheet." In truth, when you're sailing close hauled your jib sheets are already as tight as they can (or should) be. He may have been preparing for a tack, but if that were the case he would wait for that last bit of speed before easing the jib sheet.
At 35 minutes, when Emily, Richard and maid are in the living room talking about the missing sewing box and Emily goes to pick up the lighter, the mic can be seen in the top right area of the screen.