Visible crew/equipment: In the episode "where no-one has gone before" from series 1, near the start, just as Riker says "you have all the time you need", the engine consultant stops and turns around. As he does so, look above his head and you can see a microphone hanging down. The camera quickly adjusts so it's not visible any more.

Continuity: Picard addresses the ballerina as ensign, when in fact she has no pips on her uniform, making her a crewman.