Continuity mistake: As Mal approaches the desk, saying "we got a couple of DOAs" he puts his left hand directly on the corner. Cut to a shot from behind and his hand's moved about a foot.
Revealing mistake: When Simon prepares an injection for Jayne, you can see him insert it and draw back a considerable amount of fluid from the medicine vial, but the level of liquid in the vial does not drop.
Visible crew/equipment: As Wash leave the ship to get supplies, you can see a hand pulling the front of the quad. (00:17:58)
Revealing mistake: In a travelling shot, you can see an extra in the middle of the street, looking at the camera and waiting for his cue to start walking. (00:17:26)
Out of Gas - S1-E5
Continuity mistake: In the first flashback in the episode, Malcolm is showing Zoe Serenity after buying it. You can see on his left arm the stitched-up bullet hole that was from the very first episode when Mal gets shot. This episode would have taken place much later than the flashback.
Chosen answer: One of the premises of the show is that the two major Earth societies that survived, and ultimately merged, were the Chinese and Americans. As a result the common language includes much Chinese. Clothing, signage, and building designs also have a heavy Chinese influence.