The Russian Tolkien fandom has a set of very specific Fanon, mostly First Age-related, most of which was established by published big fanfics. (Yes, there is a bootleg Expanded Universe version of Arda in Russia, illegal in most of the world but legal in Motherland itself.)
The (common to most Tolkien fandoms) notion of Celegorm the blond
The notion that the Feanorians and their warriors wore a uniform of red, black and silver
Fans have ideas about female characters, especially daughters of Men, but also Elves and Hobbits. A woman always wears a dress, never breeches or trousers. If a woman is on a horse, she always rides sidesaddle, never astride. Some fans enforce this even when the woman must hike or ride for long distances.
Fans often add the title "Lord" or "Lady" to characters. Fan fiction has "Lord Erestor" and "Lord Haldir". In canon, a lord or lady is almost always the first man or woman of some place or people, like Elrond "Lord of Rivendell" (LotR Book II, Chapter 1), Denethor "Lord of Gondor" (V, 1), and Éowyn "lady of Rohan" (III, 6). An exception is Gandalf as "Lord Mithrandir" (V, 1). Some fan characters have "Lord" or "Lady" as a rank without anywhere to lord over. Lord Kinsey in Home with the Fairies lives in Minas Tirith, and might be lord of nothing but his own household. In some fan stories, "lady" is no longer a title, but a polite label for any random woman, like lady Helanthir in Troubled Waters.
Fans do not agree on whether things that appear in the earlier drafts of The Silmarillion (and its predecessor The Book of Lost Tales) are canon if they are not contradicted by the later, published, version. For instance there is a part in The Book of Lost Tales which describes how when Arda was first created, a number of creatures from outside the world fled into it; hence not everything in the world is something which the Valar put there (the most prominent example being the Faerie). This is of interest to fans because it provides a possible origin for Ungoliant and Tom Bombadil, but the fact that the passage doesn't appear in The Silmarillion casts doubt on this.
dwarfling and elfling — These words only appear in fan fiction. The suffix -ling is for "little", so an elfling is a little elf, an elf-child. Tolkien never used "dwarfling" nor "elfling", though he did use some other -ling words. So in LoTR Appendix A, a young-adult dwarf is a "stripling", and an Orc addresses a dwarf as a "beardling". Also, a hobbit is a "Halfling" (LoTR II 2), and little Ents are "Entings" (LoTR III 4). (Among fans, Elfling is also the name of a mailing list about Tolkien's languages, or elf linguistics.)
general — This army rank is fanon. Modern militaries have generals, so fans suppose that armies in Middle-earth also have them. Tolkien's armies have few ranks: Gondor and Rohan have captains, and Rohan also has marshals. The word "general" appears in LoTR IV 4, when Captain Faramir describes Boromir as "our Captain-General", which is a way to say that Boromir outranks Faramir. Boromir is a captain, not a general.
human — Tolkien's characters never use the word "human". Fans use "human" for the race of Men; but a few fans also use "human" for Dwarves, Elves, and Hobbits, because those races also have human traits.
Rohirric — This adjective for Rohan is a fan invention. Wikipedia's article for Rohirric traces this word to Robert Foster, author of The Complete Guide to Middle-earth. The word Rohirric puts the English suffix -ic on the Sindarin word Rohirrim.
Minas Tirith, the city of seven circles, is where the wealthy live closer to the castle, while the lower and outer levels attract poorer and rougher folk. This is how The Games of the Gods and A Far Green Country depicted the city. This is only a guess, but perhaps a good guess.
How short are dwarves? All measurements are fanon. Some fans think that dwarves are taller than 5 feet (152 cm). In the fan story Worlds Apart, Nymeria is 4 feet 11 inches, and the dwarves are taller than her. This might be too tall, because in canon, the secret door to Erebor is 'five feet high and three broad' (The Hobbit chapter 11), so dwarves are probably under five feet.
Fans assume that Thorin, Fili and Kili are the only dwarves in Durin's line, among the thirteen dwarves of The Hobbit. They assume wrong. LoTR Appendix A reveals that Balin, Dwalin, Óin, Glóin, Ori, Nori and Dori are in Durin's line. Bifur, Bofur and Bombur are not.
Dwarf-women have no beards. This is probably wrong: LoTR Appendix A states of dwarf-women, 'They are in voice and appearance, and in garb if they must go on a journey, so like to the dwarf-men that the eyes and ears of other peoples cannot tell them apart.' Therefore, dwarf-women have beards like dwarf-men.
Some fans believe that Elves are vegetarian, with almost no evidence. The Hobbit, chapter 8, where the Elves of Mirkwood have 'roast meats', is one of several canon references to suggest that Elves do eat meat.
In The Silmarillion Chapter 17, the Elves of Ossiriand say of Men: 'And these folk are hewers of the trees and hunters of beasts; therefore we are their unfriends, and if they will not depart then we shall inflict them in all ways that we can.' This line suggests that Elves are not 'hunters of beasts' and never eat meat, but there are other interpretations. Perhaps Elves eat meat but Men would kill too many beasts. Perhaps Elves eat fish, not beasts of land. Perhaps the Elves of Ossiriand are vegetarians but other Elves are not.
More recently, the movie of The Hobbit: An Unexpected Journey (2012) has a scene where the Elves of Rivendell serve vegetarian food to the Dwarves. But this is not in the book.
Fans are not sure how many elves have golden hair. LoTR Appendix F says of the elves, 'their locks were dark, save in the golden house of Finarfin' (or in older editions, 'house of Finrod'), but that isn't all true, because in The Silmarillion, Finarfin is the son of Indis of the Vanyar, and the Vanyar have golden hair.
Orcs in fan fiction like to ravish the daughters of Elves and Men. The canon is too vague; we know that orcs did something to Celebrían, but not exactly what. Fans suppose that they raped her. Even if orcs prefer to rape other orcs, there might be at least a few orcs who rape other races.
Aragorn was orphaned at two. This is downright contradicted by the Appendices to LoTR: Aragorn's father died when he was two, but his mother lived until he was in his seventies.
It's also very, very commonly accepted that Aragorn and Legolas knew each other well before The Fellowship of the Ring took place. It's everything but canon now. Does make sense when you consider the facts though: Elrond and Thranduil most likely keep very close contact, and so given he's Thranduil's son, this would mean Legolas has probably spent a decent amount of time in Rivendell, where Aragorn has lived most of his life up to the Fellowship.
Boromir is a He-Man Woman Hater or even a Straw Misogynist. He thinks that women can't do anything. In fan fiction, Boromir acts rude to any woman he meets; and if the woman offers to help the group, Boromir argues against her. Fans do this to make their female characters seem more special.
Elladan and Elrohir are troublemakers. Elrohir is the more sensitive of the two, and Elladan has more of a temper. None of that is in the books.
Elrond was the romantic partner of Gil-galad, the last High King. Canon has no gay romance, but leaves enough room for one. Their romance would explain why Gil-galad gave the ring Vilya to Elrond, and why Gil-galad had no children to become High King after him. Also, Gil-galad died at the end of the Second Age, and Elrond did not marry Celebrían until the Third Age. This isn't enough to prove that their love was romantic instead of just platonic.
Erestor is the grim headmaster type in Rivendell. That's not in the books. He is only "Lord Erestor" in fan fiction. Erestor in canon is "the chief" among "counsellors of Elrond's household" (LotR Book II, Chapter 2). Fans also ship Erestor with Glorfindel.
Figwit or Melpomaen is not in the books. His names are Fan Nicknames for a background character in Peter Jackson's movies of The Lord of the Rings.
Fili and Kili are caught in a Continuity Snarl about who is older. In The Hobbit chapter 8, Thorin says, 'Fili is the youngest' when Kili is also present, so Kili is the older brother. Then in LoTR Appendix A, the family tree shows Fíli born 2859 and Kíli born 2864, so Fili is the older one. Most fans accept that Fili is older, because most references copy the years from LoTR, and because Peter Jackson's movies present Kili as the younger brother. Fans have also exaggerated the age difference, so Kili has less experience and maturity than Fili; but The Hobbit happens with 2941 (in the timeline from LoTR Appendix B), so their ages are close, about 82 years for Fili and 77 for Kili.
The idea that Glorfindel of The Silmarillion is the same elf as Glorfindel of The Lord of the Rings comes from Tolkien himself, so it might be canon! Tolkien wrote in an essay (found in The Peoples Of Middle Earth) that after Glorfindel died in The Silmarillion, he came Back from the Dead before The Lord of the Rings. To reduce Continuity Snarls, some fans omit essays like that one from continuity. This allows that the two Glorfindels are two different elves.
Glorfindel of Rivendell is "Lord Glorfindel" in fan fiction. Gandalf says of Glorfindel, "He is an Elf-lord of a house of princes" (LoTR II 1), so "Lord Glorfindel" is probably correct.
Many fans think that Legolas is the eldest son of Thranduil and the next king. Legolas is an only child or has a younger sister. Canon never mentions any children of Thranduil other than Legolas, but they might still exist. A few fan stories suppose that Legolas has an older brother, which is not against canon.
The mother of Legolas (and wife of Thranduil) died before The Lord of the Rings. This idea is common, though each fan story gives a different version of her death. Fans kill the mother because they want Legolas to have more angst. The Games of the Gods, Book One, Chapter 75, defies this idea by declaring, "yep, she was still alive".
What hair color has Legolas? Peter Jackson's adaptation of The Lord of the Rings might have settled the fan arguments. He's blond. The books make only vague references to his hair color. In The Hobbit, the Elvenking has gold hair. This king is Thranduil, father of Legolas; so it is plausible that Legolas inherited gold hair. The Lord of the Rings puts that most elves have dark hair, but is vague about Legolas. Some fans still believe that movie Legolas and book Legolas have different hair colors; the authors of The Games of the Gods, I Am NOT a Mary Sue and What Grace Has Given Me defy the movie and write that Legolas has dark hair.
Morgoth, in many fan portrayals, simply looks like a bigger version of Sauron's physical form the Peter Jackson films. That's because Jackson used illustrator John Howe's Morgoth for movie Sauron. Howe was also hired for movie concept art.
Radagast has his name in the books, but never appears in them. Fans take his character traits from Peter Jackson's films of The Hobbit, where Radagast does appear.
Tauriel is not in the books. She comes from Peter Jackson's films of The Hobbit.
Thranduil is apparently an alcoholic who regularly beats Legolas senseless. This is highly unlikely, based on what we know of elves in general and Legolas in particular.