Continuity: During the last fight with the Reavers, Inara is holding a blaster gun shaped like a bow and arrow in close up shots. In any shot that shows the whole crew, the blaster changes into an actual bow and arrow. The producers didn't like the bow so they actually enhanced it with CGI to make it bigger and "cooler", so those shots of the whole crew are probably shots where they forgot to add the CGI or assumed people wouldnt notice.

Visible crew/equipment: In the closing scene where the Operative tells the Alliance soldiers to stand down, one of the soldiers (almost out of frame) lowers a set microphone.

Continuity: Watch the items on the table when Mal's giving the speech in Serenity after finding the holographic recording. There's a yellow cup, an orange object to the right of it, and a large bottle to the left. When the shot cuts to a close-up of Simon, the bottle disappears.

Other: When the camera changes shot to the security feed for the first time in River's bar brawl scene, you can see, on the top right, Jane and Mal sitting down with Fanty and Mingo and the Dancer preparing to block the camera's view. This is before the gang have even stepped into the bar.

Revealing: When the Operative is reviewing Mal's file, Mal's first name is initially spelled as "Malcom", in the next shot that has been corrected to "Malcolm".

Visible crew/equipment: In the vault scene, there's a part when the camera shifts and you can see the area outside the cylindrical vault set in the upper left of the screen.

Continuity: In the very last scene when River and the Captain are flying the ship, Mal presses a button causing it to light up. In the next shot the light is off, and in the next shot it is back on again.

Factual error: According to Joss Whedon, people in the Firefly universe speak only English and/or Chinese (see Special Features and the movie's subtitles), and yet, the writing we see everywhere is Japanese (there's a difference between Japanese and Chinese script; it's obvious for those who know).