Coming Home - S1-E13
Plot hole: The logistics of the episode don't really make sense; Amuro's home village is held by (literally) a handful of Earth soldiers cut from their main force and that spend their time getting drunk and acting as occupation force, but Zeon has a full base (again, literally, almost) next door that sends reconnaissance troops and even aircraft to check for any activity at the refugee camp. There's no reason why the undisciplined and free soldiers would stay in such a dangerous position where they could be wiped out by overwhelming forces anytime, nor why Zeon would keep a pocket of the enemy forces that they can crush with ridiculous ease.
Revealing mistake: When Lizzie awakens on the morning that Edith gets released, Edith doesn't respond and Lizzie goes to her bed to "mourn" the passing of her friend. Except that Edith is still breathing.
Continuity mistake: At the Chequers meeting in early 1944, Sir Bernard Montgomery is wearing the rank insignia of a field marshal. In the later meeting with the King in which the D-Day plans are announced, he is correctly wearing that of the lower rank of general. In fact, Montgomery was promoted from general to field marshal on 1 September 1944, nearly three months after D-Day.