Out of Gas - S1-E5
Continuity mistake: In the flashback where Mal and Zoë meet Jayne for the first time, every close-up of Mal and the one close-up of Zoë shows their hair blowing in the wind. But in the shots of them together, their hair is completely still.
The League of Villains - S3-E16
Factual error: As the villains are approaching to Earth, South America is shown just under Asia, connected to the Arabian Peninsula and the southern tip of India. The two continents are nowhere near enough to each other to be seen at the same time from space, much less connected. They are on opposite sides of the Pacific ocean, over 10,000 miles (16,000 km) apart. (00:06:45)
Here There Be Monsters - S1-E10
Plot hole: The female police officer has a theory that Johnny was the Strangler in the case the papers claimed he helped solve but this hardly makes sense. If she was researching him as was seen in the episode, surely it would have said something about Johnny being in a coma during the bulk of the murders since the story was about Johnny.
Forever Red - S1-E34
Plot hole: Tommy states that he and the other Zeo Rangers destroyed the Machine Empire's leaders and ultimately destroyed the Machine Empire except for the Generals who escaped. This is not true - Lord Zedd and Rita destroyed the Machine Empire leaders at the end of series 4, and the Space Rangers destroyed them again in Countdown to Destruction in series 6. The Zeo Rangers never destroyed the Machine Empire.
Suggested correction: How would Tommy know that? He didn't watch the show or know the ins and outs of what happened in space. This is what he believed based on his experiences on Earth. Everything you described is factually correct, but Tommy did not witness this. So, to his knowledge, he destroyed the Machine Empire.
Factual error: In the opening sequences of the show, B-17s are over the Ruhr in August 1944, and yet they're being attacked by nothing but yellow-nosed BF-109E's. The yellow nose was discontinued at about the end of the Battle of Britain, and the "E" model was discontinued even before that.