We'll Always Have Paris - S1-E24
Character mistake: When Data is on the planet, about to drop the material into the temporal rift, one of the three Datas asks which one of them should do it. The middle Data replies, "Me. It's me." Data is meant to be incapable of using a contraction.
Suggested correction: That character trait was not established until season 2. Data uses contractions multiple times in the first season.
While it was explicitly expressed in season 2 that Data can not use contractions, the fact that they did established this character trait means it could be a valid mistake since the trait is meant to show Data could never use contractions since his creation, not that he suddenly developed it some 30-odd years later. Although it would probably be best to submit it as a mistake in season 2 when it's mentioned that it contradicts what was established or at least make one character mistake for "all of season 1" than list every individual time it happened.
We'll Always Have Paris - S1-E24
Continuity mistake: The red towel Picard holds goes from draped over the front of the arm rest, to over his hand between angles.
