The SnO.C. - S2-E5
Continuity mistake: In Series 2, Episode 'The SnOC', we see Zach punching Seth with his right fist. However, when Summer gives him some ice, he covers his left fist with it (which is now bruised and swollen).
Factual error: When they first assault the city, the Greeks construct catapults. Only problem: catapults were invented at 400 - 300 BC, i.e. about 800 years later; and the first models looked similar to giant crossbows. The one-armed onager and similar types (as seen here) were a later Roman invention.
Continuity mistake: At the beginning, Kate's friend gives her ex-fiance's letter to Kate. The letter is held differently in consecutive shots.