One candle on Simon's birthday cake is lit and unlit in alternating shots.
Kaylee has just put Simon's birthday cake down on the table when the explosion happens, but in many subsequent shots of both the table and the floor in the dining room, the cake is nowhere to be seen.
The phrase "dead in the water" was used and Kaylee said they weren't moving. In deep space it's not possible to actually stop all motion. When they ran out of "gas", they did not coast to a stop as there was no friction to slow them. They would keep going at the same speed until they hit something with too much inertia to push away.
When Mal pulls the handgun out it makes a cocking or racking sound, but he's not seen racking the slide and it has no hammer to cock.
When Mal and Zoe first walk onto Serenity, a cable on the floor behind them slides across the floor in one of the shots, then appears in a different position in a later shot.
When Dr. Simon is talking with Inara about failing life support on the ship he starts to describe asphyxiation as a manner of death but low level of oxygen in the air would cause hypoxemia and hypoxia and not asphyxiation.