Character mistake: When the news broadcast on the destruction of Henrik's is shown, the chyron misspells the store's name as "Henrick's." (00:07:50)
Character mistake: When Rose thinks Mickey is dead, she says she'll have to tell his mother. But in Rise of the Cybermen in Series 2, she says that Mickey's mother has been dead since he was a child. (00:30:17)
Character mistake: The Doctor comments that the destruction of Earth will cost 5 billion lives. At the time of the episode (2006, one year on from when Rose met the Doctor), the population of Earth was around 6.5 billion. (00:25:10)
The Christmas Invasion - S2-E2
Character mistake: When Llewellyn is describing what was carried onboard Guinevere One, the UNIT computer displays the space probe's name misspelled "Guinivere" One. (00:26:50)
Character mistake: When the Doctor and Rose are walking back to the TARDIS at the end, he explains to her that Queen Victoria was hemophiliac. She wasn't. She was a carrier of the disease due to a mutation in one of her parents' reproductive cells (probably her father, Edward, Duke of Kent, because he was in his fifties when Victoria was born), but since she only had it in one of her X chromosomes, she didn't actually suffer from it as the gene for it is recessive, which is why most hemophiliacs are male. It is true that several of her children inherited the gene, and one of her sons was hemophiliac, but not Victoria herself. (00:40:55)
Character mistake: The Doctor gets excited about the major events of 1953 and mentions that everything was "off the ration" (that is, the food rationing put in place in the UK during the Second World War had ended). Rationing actually ended the following year. (00:05:55)
Character mistake: Dr Singh refers to "a single protozoa." "Protozoa" is the plural form. He ought to have said "a single protozoan." (00:08:05)
Character mistake: In an early scene with Yvonne Hartman, she says "We've just measured the ghosts' energy at 5000 gigawatts." There are at least two mistakes here: (1) watts are a unit of power, not of energy (that would be joules); (2) later on, when Hartman is showing the Magnaclamp to the Doctor, she refers to imperial tons and says "Torchwood refuses to go metric", but a gigawatt is a metric unit.
Character mistake: After the Carrionites kill Lynley, the Master of the Revels, by using the doll to make him drown on dry land, the Doctor says, while examining the body, that he's never seen a death like it. In the Third Doctor serial "The Mind of Evil", involving a monster that could make people relive their worst fears, a character died in exactly that fashion. (00:13:10)
Character mistake: David Tennant has a strong Scottish accent, and puts on the Doctor's English one. When David and Freema Agyeman are lying on the bed in the hotel, David forgets his English accent when he says, "There's something I'm missing, Martha."
Character mistake: The question to unlock one of the doors asks for the next number in the sequence 313, 331, 367. The Doctor gives the answer as 379 with a convoluted (albeit correct) explanation of happy primes, but there are much simpler solutions. The difference between the first and second numbers is 18 and the difference between the second and third is 36 (which is 2 x 18), suggesting that the difference between the third number and the next could be either 54 (= 3 x 18), giving 421 as the next number, or 72 (= 2 x 36), giving 439. It's likely that the Doctor would see that the answer is ambiguous, and yet he confidently states that the most complicated answer is correct. (00:08:45 - 00:09:15)
Suggested correction: If you know what a happy prime is, like the Doctor does, it might be simpler than your idea because you gave 2 options - 421 and 439. The idea of a pattern is there should be only one answer that fits, which would be the case if they're all happy primes.
Utopia (1) - S3-E11
Character mistake: When the Futurekind spy starts sabotaging the silo's systems, and Chantho notes that they're losing power, she says, "We're losing power-tho!" - forgetting the "Chan-" prefix that she starts anything she says with, in order to not swear by her species' standards. (00:27:00)
Character mistake: When the Doctor is taking Donna to 4.6 billion BC to watch the formation of the Earth, he says that it is the furthest back the TARDIS has ever been. But in the 5th Doctor episode "Castrovalva", the TARDIS drifted to 13.2 billion BC and the formation of the galaxy. (00:40:35)
The Time of Angels (1) - S5-E4
Character mistake: When Amy says "Doesn't anybody need me?" from the door of the dropship, there's a misspelling on the door: "Trip Hazzard" instead of "Trip Hazard." (00:16:19)
Character mistake: Idris (the human form of the TARDIS) berates the Doctor because the doors of the TARDIS open inwards, pointing to the notice on the left hand door which says 'pull to open'. Whilst it is true that police boxes did open outwards, the notice is actually referring to the telephone in the small recess behind the notice. The public are instructed to 'pull to open' the small door to access the phone. Consequently, the small door does open correctly, as in the notice. (00:23:50)