Continuity mistake: Kirk did not know who Miss Patty was. He said "I wasn't aware that you were THE Miss Patty." Later in the series, in Episode 5 of Season 6, at Miss Patty's Grand Recital, Miss Patty introduces Kirk's performance and says that he was "a talented pupil of mine". His performance was titled, "20 Years Ago Today", therefore Kirk was a student and should have known who Miss Patty was all along.
Double Date - S1-E12
Factual error: Lane wants Rory to talk to Dean about setting up Lane with his friend Todd. While asking Rory to do this, Lane asks for the date to be planned for the following Sunday, after church; however, Lane is Seventh-day Adventist and they believe the Sabbath is Saturday and therefore attend church on Saturday, not Sunday. (00:04:45)
Suggested correction: Many Seventh-Day Adventist churches do actually hold church on Sunday.
Dear Emily and Richard - S3-E13
Character mistake: Lorelai calls Rory "Suki" when they are unboxing the books Emily gave to them. (00:10:08 - 00:31:11)
Suggested correction: She calls her "sweetie."
Suggested correction: That's exactly the point of the scene. Kirk knew that her name was "Miss Patty" but he didn't know she was THE "Miss Patty" about whom everyone was speaking.
They also refer multiple times to Kirk growing up in Stars Hollow. He would have known. In Stars Hollow, there couldn't ever be another "Miss Patty." That's just not how they roll.