It's stated in Assassination Classroom that Mach 20 is "twice the speed of sound", when actually Mach 1 is the speed of sound.
One scene in Baki the Grappler involved a character who blinded people by pulling out their optic nerves... by sticking a finger into the side of his opponent's neck and pulling said nerve out. The optic nerve, which connects the eye and brain via a hole through the eye socket, has absolutely no business being there.
A Lupin III episode had a sign marking the Kansas/Washington D.C. border. Was it that hard for the writers to get a map of the United States?
Generally, Detective Conan avoids these issues by Aoyama doing research, off and on there are instances where the reasoning for even simple things in the series just don't make sense. One instance is when Ludger, a German character, appears and he speaks near-perfect Japanese. Conan deduces that his wife is British because Ludger referred to the restaurant being on the 2nd Floor, when it would be the 3rd Floor in Japanese, and that this is because the British refer to the 1st Floor of a building as the Ground Floor and then begin to count number-wise upward. The problem? Germans do that, too, so his reasoning is faulty since Ludger, being German, automatically counts floors in the German fashion, making Conan's elaborate reason why his wife must be British obsolete.
Code Geass loves its chess metaphors and concepts. Lelouch is also an avid chess player throughout the series, which he displays in his strategies. However, he always plays by moving his king first, because he believes the king should lead by example. In chess, it is not possible to move the king on its first turn; it only can move one space in any direction, and is blocked in by its own pieces. It's also an extremely risky way to play chess in general. Also in the second season, Schneizel (who is supposed to be an even more awesome chess player than Lelouch), stalemates Lelouch by moving his own king into check, which is a completely illegal move, as any chess player knows.
Black Butlersubverts this trope at the beginning of episode 4. The episode starts out with Sebastian and Madame Red playing chess, and he stalemates Madame Red by moving his own king into check, which is a completely illegal move, as any chess player knows. In response, Madame Red angrily (and correctly) yells to Sebastian that he just made an illegal move. In turn, Ciel poetically states that many people make this move in real life, as no matter how prepared a king is, unfortunate circumstances can allow even a pawn to strike him down.
He is, of course, referring to the person who started the fire that killed his parents.
If you didn't know Osamu Tezuka was a certified M.D before he was a mangaka, you'd think he was this. As it is, the official translation actually has footnotes explaining the discrepancies between the manga and real life, with some being honest mistakes like claiming that dingoes are descendants of dogs brought to Australia by Europeans when in fact they were already present before colonization, which is a mistake anyone could make at first. Blatant errors in medical knowledge, are actually embellishments done on purpose for Rule of Fun and Rule of Cool, and a situation where a woman gets her ovaries removed and later on appears as a man is a case of cross-dressing, which was been mistaken for a literal Gender Bender due to the use of phrases like not being able live like a woman anymore used in dialog, making this a case where the readers, reviewers and critics are the ones that committed the failure in research, because a medical doctor like Tezuka would know that getting their ovaries removed wouldn't cause a woman to become a man.
Yoshiyuki Tomino named the fascist adversaries of Mobile Suit Gundam after the Nazi-like faction in the original Star Trek episode "Patterns of Force". He apparently didn't go back and watch the episode when he made this decision, because in that episode Zeon were the people being oppressed, and the pseudo-Nazis were actually called Ekos.
Electric-types used to damage Ground-types in the past, which is a No-Sell fact in the games. Ironically, once Pikachu damaged a Rhydon with Thunderbolt by striking its horn, later on when abilities were introduced said horn turned into the Lightning Rod ability for the Rhyhorn line, which draws and nullifies all Electric attacks.
Several attacks that shouldn't affect Ghost-types still affect them anyway. This is more of a random occurrence. The inverse has also been falsely claimed as true, especially for physical moves.
Ash's Sliggoo evolved during its own Rain Dance. In the games, this is impossible to do, as the rain has to be natural rain that isn't started by Rain Dance.
At least in the dub, Lillie erroneously refers to the item Potion as "Lotion."
Brock once claimed that Water-types are weak against Fire-types. Besides not making sense, anyone who's played one of the game for even a few minutes knows it's the reverse.
A Wind Named Amnesia: In the film, Little John is seen with an LAPD Sheriff badge. However, the LAPD (Los Angeles Police Department) has no Sheriff, they have a Chief of Police. The Sheriff is part of the LASD (Los Angeles County Sheriff's Department).
Lord Marksman and Vanadis: In one episode of the anime, Tigre dry fires a bow (pulls the string and lets go without an arrow). Any self respecting archer, especially an archery nut like Tigre, should know that you should never dry fire a bow, as without an arrow to absorb the energy, the bow could crack apart.
In one episode of Sailor Moon, the Dark Kingdom tries to find Sailor Moon by installing hair steamers with DNA scanners. The scanner picks up on the one strand of Usagi's hair in Minako's hair (via transferrance from her brush) leading the youma there attacking her. The problem with that is hair contains no DNA; hair follicles do, but the likelihood of Minako pulling a follicle out simply by brushing Usagi's hair is unlikely.
Mother's Basement made a whole video on why Sword Art Online would be the worst game in history. Among his complaints are that the games are incredibly unbalanced, a game of that scope that sold just 10 000 on its opening day would be a massive financial flop, and how the anime claims that no one uses swords or snipers in Gungale online because of how hard they are to use, ignoring the fact that most gamers would pick those weapons solely because of how hard they are to use (and how ludicrously OP they are if you know how to use them).
In Episode 5 of Neon Genesis Evangelion, Dr. Ritsuko Akagi states that "Angels are composed of matter that displays the properties of both particles and waves, just like light". All particles have wave-particle-duality, not just photons, including the particles that compose boring baryonic matter.
During the Black & White arc of Pokémon, Ash and his friends meet Cameron, a Unova league hopeful who says that he has all seven badges he needs to compete — except, as our heroes point out, he need eight badges. He also believes that the Unova League tournament will be held in Ecruteak City — which is in the Johto region. Oh, and when he finally gets to face Ash in a full battle in said tournament, he believes that "full battle" means "5-vs-5". It's actually 6-vs-6, meaning that he inadvertently handicaps himself.
An unusual example in the Ranma ½ manga. Gosunkugi is searching for Ranma's weakness and the manga's supposedly omniscient narrator takes the reader through a montage of what the Tendo residents have to say, including "The Tendo Family Pets". Only Soun protests that he is not a pet, but the panda and pig are Genma Saotome and Ryoga Hibiki respectively in their cursed forms. Literally no one in the panel is a pet.