A lot of details went into the researching behind the locations, schools and universities that appear in the Tsubasa Reservoir Chroniclereal-world AUFreefall, as well as the symptoms of leukaemia, which a main character is diagnosed with early in the plotline.
Between Minds (Portal and Half-Life) is packed with this with everything from helicopter specifications to language translations to first aid to building layouts to real-life weapons to cut game material to the actual mechanics of an EMP.
The author of Hogyoku ex Machina asked her high school readers if they could define tuberculosis without using Wikipedia to determine if Ichigo and friends would reasonably know. She even checked if House airs in Japan to justify Ishida watching the show.
Child of the Storm takes it up a few notches and shows that this part of Nimbus Llewelyn'sSignature Style. The story uses historically accurate Norse titles and foods, drawing on a lot of minor throwaway details from obscure corners of each of the canons and displaying an awareness of basic physics (in chapter 54, Harry is flying on a souped up Firebolt, which, in the interests of fairness, can't exceed its production top speed on the Quidditch Pitch. Harry considers that he'd rather not find out what would happen to him and the broom if he suddenly lost a third of his speed, and skirts the Quidditch pitch).
It also displays awareness of certain political realities - for instance, it is abundantly clear in earlier chapters that Harry feels that he has a very large bone to pick with Odin, and it is repeatedly pointed out to him that Odin is an absolute ruler and cannot/will not accept such levels of defiance in public, even from his grandson. In private, under certain circumstances, he is rather more accommodating and understanding, the implications of Younger than She Looks where an attractive teenage girl is concerned and uses the exploration of consequences of the basic premise (Harry is the son of Thor, who was incarnated as James Potter in his first shot at humility) of the fic as the engine of the plot.
In Equestria: A History Revealed, the author clearly has done some research into the show and the franchise as a whole. Little in-jokes from the fanbase are prevalent in the fic, and most impressively, the author manages to tie in even the most obscure sources to create one unified Equestrian history canon, pulling from the show, comics, and even the often-forgotten chapter books.
In the My Little Pony: Friendship Is Magic fanfiction Silent Knight, Nightfyre's flightsuit puts constant pressure on her limbs, ensuring proper blood-flow to the brain during stressful maneuvers. This is the same way real life flightsuits work for jet and fighter pilots.
In the same chapter, Ash Coat notes some materials on the human sword that genuinely act as absurdly sharp material when treated correctly; it's suggested that some Applied Phlebotinum is employed to sustain this condition.
Cornova did a lot of research with regards to the ins and outs of the Pokémon anime for Poké Wars.
A prime example is how he included an entire chapter on Kaiza, a character from an episode of Pokémon that never made it out of Japan.
Also, numerous one-shot characters (Duplica, Yas and Kas Gyms, AJ, etc.) are portrayed perfectly in character.
The author behind XSGCOM has an author's notes section after every chapter, where he/she explains from exactly where in the original works every character, plot-twist and gadget in the chapter came. He does the same for Hunted Tribes and New California Dreaming.
In his first story, Legacy, this is most apparent in the details related to the victim's wound. His later stories have substantial Notes sections that explain the more obscure allusions, among other things.
His Season 1 reimagining story, The Legend of Total Drama Island tends to show his work in-story for such things as inventive solutions to certain problems, and the effects and treatment of hypothermia.
One of the reasons the Homestuck fan blog Brainbent is so popular with fans is because of how accurately and sympathetically it portrays people with mental health issues (except John, who's disorder is unspecified). In particular, Autistic readers really like the non-stereotyped portrayal of Autism.
Friendship Is Optimal has a lot of accurate science, and the author has clearly done a lot of research on artificial intelligence and philosophy.
As part of its Deconstructor Fleet, quite a lot of the setting of Sonic X: Dark Chaos was based on (and inspired by) obscure Bible stories and passages, and so is much of the symbolism. A lot of the Emirate of Mecca was based on the actual Hadiths of Muhammed as well. Several lines of dialogue from Angel characters are in accurate Hebrew and Latin too.
Sonya's plasma-focused katana is correctly depicted as a backup weapon. In mediaeval Japan, the samurai preferred bows and spears over their swords, which they used if anything got too close. She also prefers using guns - the samurai admired firearms and used them extensively after they were introduced to Japan.
Maledict's character is a personification of Machiavelli's eponymous Prince - wanting to be both feared and loved - rather than the Popcultural Osmosis version. It helps that the author has actually read The Prince in real life.
One of Jesus' military assistants is Lucifer. This seems rather unusual considering Maledict's presence, unless you know that Lucifer and Satan are actually different biblical beings entirely.
Every hotel, restaurant, new age shop, and strip joint mentioned in Origin Story is an actual location, and all the prices involved are authentic. The writer mentions several times in his notes that he actually called these people up and asked them questions. Of special note were the women who worked in the men's department at Macy's in New York City, who actually had a twenty minute long phone call with the writer in which they described what could be seen outside the store from the window's of their department.
Breech Loader, author of fanfiction Prison Island Break, has put considerable research into such things as medical procedures, prison etiquette, blood types, speech impediments, and the kinds of injuries that would really be caused by the weapons used in the story.
The story is also commended for maintaining the personalities of the canon characters even with regard to other canon characters they'd never interact with in canon.
"Celestia Sleeps In" and its sequel "Onto The Pony Planet" by Admiral Biscuit (link) has a blog post for every chapter detailing all the research the author has put into it. This ranges from historical 18th century technology to social mores to anthropology to physics to even actually preparing Equestrian foods in real life to see how the author's human digestive system handles them.
The crossover Shepherd Of The Stars features a lot of Codex entries where the author shows off how much work went into researching the different settings. The author generated randomized tech trees for each faction in the story, tracks what technologies they have researched, and even calculated the composition, armaments, and construction costs of the various fleets that are featured. Considering there are currently ten different factions, this translates to an impressive amount of work.
The Axis Powers Hetalia fanfiction Awakening opens with the author admitting to having "a morbid fascination with this period and a deep respect for history" and thus writing the fic as realistically as possible. Various chapters end with notes explaining Josef Mengele's experiments on twins during World War 2, how the twins were cared for, and what happened to most of them after.
The Zootopia story At Our Core has the ZPD follow proper police procedure due to the writer coming from a family of police officers.
In Vathara's work Embers, the Author's Notes go a long way in explaining why certain things happened the way they did using real life comparisons including the cultures of Arctic tribes, properly translated Chinese names and terms and the psychology behind a character's portrayal.
The Kirby fanfic Wings of Change goes into a fair amount of detail about wing structures, such as having minute receptors on the wings and a large collection of bones, justifying how Meta Knight (and later Kirby) can fly, with the species' powerful lungs also helping them out. It also discusses the actual processes of flying such as needing a high lift-to-drag ratio which reduces the amount of energy needed for sustained flight.