Tolstoy, specifically War and Peace. He's one of the big names in Russian Literature and the quintessential example of tl;dr (too long, didn't read). However, if your character is a Book Worm and a member of Genius Book Club, they will know him in-depth. Perhaps they will throw a quote here and there.
Ernest Hemingway: Mostly his biography is referenced. His works, as great as they are, play a second fiddle to his captivating personality, real life adventures and utterly tragic fate. People seem to know as general knowledge that he got the Nobel Prize in Literature. If his writing does get mentioned, it's invariably his short novella The Old Man and the Sea.
Mark Twain: It's known that his name is a pen name. He was named Samuel Clemens. It's known that he was a Deadpan Snarker and his quotes are quite favourite. His most referenced books are The Adventures of Tom Sawyer and Adventures of Huckleberry Finn. Some of his scenes are a staple of the Americana, like the conning of another kid into painting a fence in Tom Sawyer (rarely will anyone mention Tom's other pranks, such as feeding the cat castor oil). Huckleberry Finn is unfortunately most well known for liberal use of that word; not many people seem to remember, say, Huck disguising himself as a girl or one of the con artists appearing naked on stage with stripes painted on his body.
Jane Austen: If a character is a teenage girl or a woman in her twenties who is fond of reading, she will read Jane Austen. Especially if the work is a chick flick, and double-especially if it's Clueless. If boys and men like her, they might be made fun of for it and their manliness might get questioned. At times her novels will be bashed because she only ever wrote about girls getting married. Others might (mistakenly) remember that she wrote something about zombies.
The Brontë Sisters: If your character is a girl or a woman and a Book Worm, and if she's somewhat Darker and Edgier that a Jane Austen reader, she will read Jane Eyre or Wuthering Heights. You might have heard that those crazy but talented sisters wrote other books as well, but they are not as noteworthy as these two Gothic novels.
The only Turkish novel is Mehmet My Hawk by Yashar Kemal.
The only Lebanese author is Khalil Gibran.
The only Israeli novelist is Amos Oz.
Enid Blyton's settings were forever 1930, even though she was writing well into the 1960s.
Edgar "Allen" Poe wrote only gothic horror, including "The Raven", "The Pit and the Pendulum", and... that's about it. Spoofed for instance in Sabrina the Teenage Witch when the aunts summon Poe's spirit to read them something spooky for Halloween but he's only interested in sharing romantic poems.
If someone says "I know Spain" on the Internet, nine out of ten times it will be followed by "I read Homage to Catalonia".
In one of the more egregious mistakes in Angels and Demons (which is said to have an errata list longer than the actual book, but that's for another article), Dan Brown has a so-called "British" journalist (in a fantasy about achieving success in the near future) liken himself to Dan Rather — despite Rather being unknown in Britain. A real British journalist, indulging in such a fantasy, would liken himself to Jeremy Paxman or Sir Trevor McDonald — who, unsurprisingly, are just as unknown in the USA.
If there's a mention of any character from Arthurian legend, it's almost always going to be Arthur, Merlin, Lancelot, Guinevere or Galahad. And where did the Arthurian legend come from? It's always England—never Wales or France, where much of the story is derived from. Arthur is usually portrayed as English even though he was a Briton which, at the time, meant what we now call Welsh. At the time of the Arthurian legends, the English were a bunch of hired mercenaries brought-over from Germany who were starting to cause trouble. Here as elsewhere, there is no difference between the creation of a legend and the content of the legend.
The only Bible stories you're likely to see referenced are Adam and Eve, the Great Flood, the Exodus, David and Goliath, and the story of Jesus.
The list of American folk heroes is a pretty short one: Paul Bunyan, Pecos Bill, John Henry. They tend to be outnumbered by actual historical figures who have simply had amazing feats attributed to them: George Washington, Davy Crockett, Jim Bowie, Hiawatha, Casey Jones.
The only non-Western fairy tales most people have ever heard of are from The 1001 Nights (often mistitled The Arabian Nights): "Aladdin and His Wonderful Lamp", "Ali Baba and the Forty Thieves", and "Sinbad the Sailor", African, Native American, Polynesian, Chinese, Japanese, and (to a lesser extent) Russian folktales tend to be known only to scholars (or to people familiar with folklorist Joseph Campbell). Disney tried to remedy this in the late '90s by for the first time choosing subject matter from a non-European or non-Semitic civilization: Mulan. (The Lion King, while ostensibly African, was just a variation of Hamlet.)
Ancient mythology consists solely of Greek, Egyptian, and Norse. And even then, you're not likely to hear of any of the more obscure myths or deities.
The Greek and Roman deity names are often mixed up. The same extends to heroes and demigods: Hercules is almost never called by his Greek name Herakles, even when everyone else is being called by their Greek name. The hero of The Odyssey is Ulysses.
The Iliad and the Odyssey are the only works dealing with the Trojan War and between them they cover the whole war from inception to aftermath. There is no such thing as The Trojan Cycle.
Inferno is the only thing Dante Alighieri ever wrote, and it's certainly not part of a larger work. Alternately, he only wrote The Divine Comedy. Sadly, almost all media based on the poetry of Dante Alighieri is based on Inferno, the first part of the Divina Commedia (aka The Divine Comedy). Few people are aware there are three parts, the others being Purgatorio and Paradiso. Even fewer people outside of Italy know that he wrote other stuff too.
He just might have said something about plans going wrong.
And a thing about a haggis.
He is the only Scottish poet.
"The Listeners" by Walter de la Mare. Most Brits of your parent's era can recite this off by heart.
"Timothy Winters" by Charles Causley.
French poets? Don't bother to look further than La Fontaine, Apollinaire, Verlaine or Rimbaud. (The only actual French poem anyone can recite is probably "Alouette" or "Frere Jacques", and they're, more technically, songs.)
Irish poets consist of William Butler Yeats, and he only wrote one poem, "The Second Coming." And the only part of that poem anyone is likely to recognize is "Things fall apart; The centre cannot hold." And if his name is mentioned, it's better than even money it will be pronounced "yeets" (as opposed to "yates").
Exceptions to this trope from This Trope in Literature:
In a lot of Middle Ages, Renaissance, and English Renaissance literature, it is pretty clear that many writers thought (or thought that their audience would think) that every non-Christian religion worshiped the Greek or Roman pantheon. The Song of Roland portrays Muslims as Apollo or possibly Apollyon worshippers — the actual word in the poem is Apollin, while Shakespeare had characters reference the Greek gods in stories that supposedly took place before those gods' introduction to the specific settings (although that was a method of Getting Crap Past the Radar, since swearing upon the Christian God was illegal even if onstage).[[/note]] In A Midsummer Night's Dream, despite being set in Greece, the Roman pantheon is used. Though since most of the mythology came from translated Latin work which was either passed down in monasteries or preserved by Muslims (and in that case translated as many as three times), the usage is understandable.
Harry Potter and the Philosopher's Stone has the real-life alchemist Nicolas Flamel as an important character, alongside his theoretical creation, the Philosopher's Stone, and many people didn't realize he was based on a real person. Rowling even got the colour of the stone (red) correct. That applies only to the original, for the American edition has the title changed to Harry Potter and the Sorcerer's Stone. She also included mythological beings such as hippogriffs (griffin/mare hybrids) and basilisks that some readers thought were original. Those who didn't probably knew those creatures from Dungeons & Dragons. In Harry Potter and the Goblet of Fire, Rowling introduced veela, which many thought she invented; in fact she drew them from East European (particularly Polish and Serbian) folklore. In fact, most of the creatures Rowling included in her novels were drawn from mythology; she only invented a few of them.
Terry Pratchett manages to invert the trope quite regularly in the Discworld series, by drawing inspirations from obscure Roundworld phenomena and essentially migrating some into the Discworld wholesale (vampire watermelons in Carpe Jugulum, The Glooper in Making Money, etc.). It helps that there's a book out (The Folklore of the Discworld) that points out some of the allusions, with many others it might take several readthroughs to notice. Long before the book was published, explanations of the in-jokes in the books were present in L-space. Many have been amazed by the number of annotations there - and the fact that the jokes frequently work at least partly for those that don't recognise the more obscure British references.
Ursula Vernon's Digger is also good with this, usually involving tidbits that look like fiction unless you look it up. In addition to a use of the previously mentioned vampiric gourds, large swaths of the hyena tribe's mythology revolves around their tremendously high infant mortality rate—something they share with real life hyenas.
The Brief Wondrous Life of Oscar Wao manages to avert this at every possible turn without treading into Genius Bonus territory. The annotations are there to help with the things most people wouldn't get but occasionally references are made to concepts that requires fairly in depth knowledge of geekdom which the narrator assumes the reader will understand.
Piers Anthony likes to include obscure references in the Xanth books. One such, in Crewel Lye: A Caustic Yarn was a reference to Langston Hughes' poem Fog (which no one can recite beyond "The fog comes in on little cat feet..").
P. G. Wodehouse, despite writing light-hearted, comedic novels, averts this. Apart from the many quotes from Shakespeare and the Bible, there are also references to, for example Tennyson, Omar Khayyám, Longfellow, Walter Scott, Laurence Hope, William Butler Yeats or Thomas Moore - in just one novel (Summer Moonshine). Jeeves's favorite philosopher is Spinoza, and he quotes, among others, Marcus Aurelius.
Reginald Hill, especially in the Dalziel and Pascoe novels references a whole range of English literature.
Israeli writer and Nobel Literature Prize laureate S. Y. Agnon is particularly notable for using a lot of references to Biblical stories and Jewish religious writing which anyone who isn’t well-versed in Judaism won’t recognise. These references are used either as metaphors (missing those could cause one to miss a key point in the story altogether) or as somewhat obscure references, to the point he practically invented a new language based on archaic Hebrew and the occasional neologism.
Israeli poet Natan Alterman was also a very hard averter of this: The Alterman Notes, vol. 3 mentions he had a vast and diverse library, in six languages and on many subjects, which he used as reference material for his work. He, however, makes these references much more subtly.
British statesman Lord Chesterfield mentioned this phenomenon in Letters to His Son: "Frivolous, futile people, who make at least three parts in four of mankind, only desire to see and hear what their frivolous and futile precursors have seen and heard: as St. Peter's, The Pope, and High Mass, at Rome; Notre Dame, Versailles, the French King, and the French Comedy, in France." (letter 148)
The Lord of the Rings has got to be the most extreme inversion of this. The books are littered with references to the (at the time) unpublished wider mythology that only Tolkien would understand. Readers would have to wait until the publication of The Silmarilion over 20 years later to understand the meaning of names like Númenor, Beren, Lúthien, Undying Lands, Morgoth, Ungoliant, Fëanor, etc.