When Zelda is arguing with Hilda about cooking the dinner she has her hands on her sides but when Hilda points her finger at the table her hands are crossed in front of her.
Harvey Kinkle says that his sister helped him pick out Sabrina's gift however, in a later series when Una Kinkle is born, she is said to be his only sister.
When Sabrina returns with Salem she gets downstairs but enters a room then appears walking down the stairs.
In the sequence in which Sabrina and her aunts go door to door looking for Salem, every door they go to is the exact same one (notice the same photo on the wall in the background inside). They just flipped all of the shots with the door opening on the right to make it look like different doors. It's especially noticeable in the shot of the guy dressed like a Christmas tree because the house numbers on the left side of the screen are flipped.
Sabrina is a witch and uses her magic to change clothes and conjure up completely new outfits all of the time, even in the title sequence. But when Salem the cat tears up her favorite Christmas sweater, not only does she fail to use her magic to repair it but even goes upstairs to change the old-fashioned way.