Deliberate mistake: When the Doctor begins to "fade" in the transporter room his mobile emitter fades with him. Since it's made of solid matter and is not a hologram, this shouldn't be possible.
Other mistake: The Sliders discover the reverse-time issue when they notice that the clock on the wall is backward. However, in other shots, clocks look normal. Look carefully at the clocks on the judge's desk as the camera pans by.
Well Bargained and Done - S1-E2
Character mistake: Tybus says he has a lance of 5 mechs. In Battletech, a lance is 4 mechs. While a Clan star does have 5 mechs, Tybus is an Inner Sphere character.
Stay Tuned - S1-E17
Continuity mistake: When Tom and the English teacher walk into the campaign HQ, a shot is shown of a security guard glancing up the stairs, and a blonde woman carrying a poster walking past. After Tom and the teacher have finished having a conversation with a balding man wearing glasses, the exact same shot is shown again.