The War Without, the War Within - S1-E14
Other mistake: In dialogue, Starbase 1 is described as "100 AU from Earth," which is over three times the orbital distance of Neptune, but still within the Solar System. But the onscreen graphic shows the starbase next to a planet that can be visually recognized as Earth; portions of North America, including the Florida peninsula and the Gulf of St. Lawrence, are visible.
Other mistake: There is a countdown on the movie screen. It is then reflected in Dusty's glasses. But the numbers are not reversed in the glasses. (00:41:00)
Other mistake: This comes up in other sci-fi scenarios too - the shuttle is heading down to the planet's surface through heavy cloud cover/disturbances. Everyone acknowledges it'll be a bumpy ride, and the flight down is very fast and tense, trying to keep track of escorts, etc. But there's no need for it to be a fast entry. Their ships aren't aeroplanes which have a minimum speed and have to keep moving forwards. They could just hover and lower slowly through the clouds at a safe speed.
Other mistake: The kids are viewing the VHS tape...but somehow the tape ends up in the old Hispanic woman's house...who then shoots it.