Factual error: Looking at a nuclear explosion that close will burn your retinas irreparably. One can be blind 30 miles away, just by seeing the flash. The little girl at the start is looking directly at the explosion when it hits, and continues to watch.
Factual error: At the very start, the lead parachutist says they're jumping from 30,000 feet, the altimeter confirms it, everyone whoops and cheers, and he gets them to put their masks on just before jumping. Two problems - the plane they're in is a De Havilland DHC-6 Twin Otter with a maximum altitude of 25,000 feet. And at 30,000 feet in an unpressurised aircraft there'd be nowhere near enough oxygen for shouting and cheering - they all should have had their masks on already.
Lost/Found - S1-E1
Factual error: A mechanic is outside the ship repairing the shields. A fire starts, and the fire burns just as it would in an atmosphere. Impossible in the vacuum of space.
And in the Beginning... - S1-E1
Factual error: The show is set in 1991, and Dexter walks past a Lehman Brothers stall at the jobs fair. The problem is that the stall and the marketing materials on it have the modern-day Lehman Brothers logo and branding. Back in 1991, the bank was called Shearson Lehman Brothers and had a completely different logo and colour scheme. It was only spun off into Lehman Brothers in 1994. (00:10:03)
Factual error: At the tanker / airplane accident site, Tommy states, "When 11,000 gallons of crude oil hits 100 octane AVGAS - it tends to go boom." The problem is there was no AVGAS present at that crash site. The plane involved in the accident was a King Air - which has turbine (jet) engines, not piston engines that would require AVGAS. A King Air is fueled with 'Jet-A' fuel. (00:28:00)
Factual error: Alex Peterson is continually referred to by everyone as the Secretary of Defence. This is an American title. The British equivalent is Secretary of State for Defence. "Defence Secretary" is used as an abbreviation; Secretary of Defence never is.
Episode #1.5 - S1-E5
Factual error: A VHS of the movie Indecent Proposal can be seen on the shelf in Dexter's bedroom. This episode takes place on 15 July 1992, but Indecent Proposal didn't release until 1993. (00:11:06)
Factual error: Originally described as a subaltern (the correct equivalent rank to lieutenant in the ATS), Sandra Wesgate is later promoted to captain, which is incorrect as the ATS equivalent was junior commander.
Factual error: Abby gets out of the train via the driver's door. You see a front view of the train. The number on the train is 321434. This train runs on overhead lines, and it is being shown as a Gatwick Express train from Gatwick Airport to Victoria. The railway from Gatwick to Victoria has a third rail conductor, so this Class 321 train would not have been able to move. (00:50:02)
Episode #1.6 - S1-E6
Factual error: In Episode 6, Vincent comes out of his apartment building and before he reaches the curb, and has a chance to hail a cab, one is waiting for him. There was no ability to book a yellow cab in the 1980s.
Factual error: Delores strikes a match and there is a huge explosion like a fuel-air weapon. The room has been filled with more than the usual amount of oxygen, but oxygen does not burn by itself; it combines with fuel, and that is the fire. The match would burn much more intensely and quickly because of the extra oxygen, but the only fuel would be the match, the cigarette, and perhaps the rest of the box of matches, which would only make a small fire.