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.
The Message - S1-E15
Continuity mistake: At the beginning of the episode, when Jayne reads in the note from his mother that she enclosed something in the box for him, there is a shot of Zoë reaching for a crowbar to open Tracey's coffin. Jayne's left hand is visible on the far right of this shot, holding the hat that his mother knitted for him. In the following shot he takes the hat out of the box despite already having it in his left hand in the previous shot.
The Message - S1-E15
Continuity mistake: At the beginning of the episode, Jayne receives a hat from his mother in the post, and when he takes it out of the package it is covered in wood shavings or some similar packing material. He puts it on his head like that, but for the remainder of the scene, the hat alternates from being covered in shavings to clean and back again.
The Message - S1-E15
Other mistake: The characters in this show often use the made-up curse words 'gorramn' and 'gorrammit.' Toward the end of this episode, just as Tracey grabs Kaylee as a hostage, he shouts, "Gorrammit, Mal." The subtitles read: "You're a rabbit, Mal." (00:35:50)
The Message - S1-E15
Continuity mistake: The cut the doctor makes on Tracey is small, but once Mal is on top of Tracey, the cut is a lot longer.
The Message - S1-E15
Continuity mistake: In the first scene in which the crew is back on Serenity, they are discussing Mal and Zoe's recent "delivery." Jayne is shown taking his hat off. Two shots later Jayne is shown in the background with the hat still on his head. Six shots later Jayne is shown with his hat in his hand again.
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.