Other mistake: With power out Magnum somehow lights the gas stove with the knob, which is electric. Should have used a match. (00:26:40)
Galactica Discovers Earth: Part 1 - S1-E1
Revealing mistake: When Troy and Dillon land on Earth, they take their bikes onto the freeway. They're followed by a large group of curious bikers. When two of the bikers move alongside Troy and Dillon and start talking to them, it's obvious that all four bikes are now mounted on a trailer, instead of riding on the road. Easiest way to tell is that their heads are higher up than the other bikers and the front wheel isn't turning.
Deliberate mistake: In the scene where the ministers are all crammed into the train carriage if you look over Jim's shoulder there is a sign ("Notice" "In case of Fire" etc), while the headers are in English the actually paragraphs are written in gibberish.
Who Was That Baby I Saw You With? - S3-E7
Character mistake: After the mixup where a baby's grandmother took baby Andrew instead of her own grandson, in trying to track her down, Muriel mentions how since it was the baby's grandmother (instead of a parent), it would be hard to track her down because her last name would be different from her grandson's. This is not fully correct. Generally a baby will have a different last name than its MATERNAL grandmother, but the same last name of its PATERNAL grandmother, so Muriel's comment was only half accurate.
Episode #1.1 - S1-E1
Character mistake: In the opening minutes of the first episode, as the Erasmus is struggling through a storm, Blackthorne is told that the boatswain is dead. He checks the corpse and then says, "Get the body aloft," which would entail carrying the deceased up the mast. The correct order would be "Get the body topside."