Bushwhacked - S1-E2
Visible crew/equipment: As Simon boards the derelict ship, a handheld camera is reflected in his visor.
Bushwhacked - S1-E2
Visible crew/equipment: As Zoë and Mal investigate the derelict ship, stage lights are constantly reflected in both their visors.
Visible crew/equipment: In the celebration at the beginning, the shot where Jayne receives the rain stick, a camera crew is briefly visible just to the right of Jayne's head right before he says "I will treasure this."
Jaynestown - S1-E4
Visible crew/equipment: When Simon poses as a buyer and gets a muddy hand print on the back of his suit, he and the gang are given a tour of the mud complex. In the first shot, the shadow of a camera can be seen on the ground in the very bottom left corner of the screen.
Visible crew/equipment: Right after Kaylee tells Mal that she's not speaking to him there is a wideshot of the ballroom dancers. In the archway at the back of the room Joss Whedon, the director/creator, walks into shot wearing a Hawaiian shirt, shorts and a baseball cap with headphones around his neck. Just as he raises his hands to speak to the dancers the shot cuts to Inara and Atherton.
Visible crew/equipment: When the fake ambulance that the crew made to infiltrate St. Mary's hospital lands at the main entrance, it lands on flat foot pads. But in the following shots the casters that were used to move the prop around the set are visible beneath the landing feet.
Visible crew/equipment: As the camera goes down through the ship to River standing on the catwalks in the cargo bay (eavesdropping on the crew as Kaylee tells them about how she killed the 3 guards on Niska's ship in War Stories), look at the kitchen table on the far right of the screen. There is a man with white hair bent over the table reading a book. He isn't part of the ship's crew, more likely part of the film crew.
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)
Visible crew/equipment: As Serenity touches down on the Eavesdown Docks on Persephone, a crew member can be seen pushing Wash's mule down the ramp just after Mal talks to Kaylee about passengers and the ship's Compression Coil.
Visible crew/equipment: When Wash tells Zoë to be careful as they touch down on the Eavesdown Docks on Persephone, a hand can been seen pulling the Mule down Serenity's ramp in the bottom right corner of the shot as Wash leaves.
Visible crew/equipment: When Simon asks Mal why he is so scared of the Alliance, Mal punches him. On the bottom left side of the screen, someone's hands are visible catching Simon as he falls.
The Message - S1-E15
Visible crew/equipment: Tracey threatens the crew with a gun on the bridge when he hears that they're about to call the cops. He says that he'll blow a hole right through Wash if they do. There is a shot of Mal saying "You mailed your ugly business to Zoë and me, Tracey." As he says this, a crew member's hand can be seen above them (through Serenity's windshield on the left of the screen) grabbing one of the pieces of fallen ice.
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.