Incorrectly regarded as goofs: A man is wearing a cross in the opening act. He is a priest; not every man in town is Jewish.
Continuity: The sun angle changes several times during the matchmaker song sequence. Most obvious when Chava is pushed onto a sunny pile of hay that was in the shadows seconds before.
Continuity: Before the Sabbath, the candles are shown on the table, lit. Soon after, we see Golde lighting them after saying the blessing.
Audio/visual unsynchronized: Near the end when the baby is crying, his mouth isn't moving.
Revealing mistakes: During the dream sequence, the wires holding up Fruma Sarah while she is "flying" are visible.
Continuity: During the first few shots of crowd holding candles on their way to the wedding, the sun changes positions above the horizon.