Remembrance - S1-E1
Other mistake: Picard gives the mysterious girl some Earl Grey. She has it in hand only in one shot, the faraway one when the two Romulan servants are leaving. The way the scene goes, at no point during the whole scene does she have time to even take a sip. (00:19:05)
Other mistake: Paul hands Abby a very thin envelope. It is supposed to contain: an insurance policy, a car title, a property deed, and an airline ticket, among other items. But that is impossible given how thin the envelope was.
Episode #1.7 - S1-E7
Other mistake: Surveillance cameras are shown on monitors when the mass panic kicks in even though any chip-dependent electronics shouldn't work in the game world (previously stated by Chotu). (00:08:20)