I'll start with: In the Cemetery Where Al Jolson is Buried by Amy Hempel http://fictionaut.com/stories/amy-hempel/in-the-cemetery-where-al-jolson-is-buried.pdf Read it a few years ago and it keeps coming to mind.
Understand, by Ted Chiang. "Tower of Babylon" and "Story Of Your Life" are also two excellent ones he's done.
These are among my favorites, even if I wouldn't consider all seven of them to be the best work put out by their respective authors. I could write a three page book report on each one... but... I think I'll try to keep this post on the slightly shorter side. It's hard for me to describe why I like most of these the way I do without simply spoiling everything about them. The Lottery by Shirley Jackson Fair Extension by Stephen King Sweat by Zora Hurston The Phoenix on the Sword by Robert E. Howard A Small, Good Thing by Raymond Carver The Yattering and Jack by Clive Barker The Dunwich Horror by H.P. Lovecraft
I was a Russian language major in college on top of my computer science degree, and although my Russian has mostly gone, I was once good enough at it to read The Queen of Spades and The Shot (both Pushkin) in the original language. Those are objectively good short stories, but their status as my favorites may have to do with how cool it was that I didn't need a translator. Also, All the Myriad Ways and Night on Mispec Moor (Niven).
"Shoggoths in Bloom" by Elizabeth Bear, for being a glorious, beautiful rumination on freedom and choice that also deconstructs Lovecraft racism. "A Hole to China" by Catherynne Valente, for being a vivid, slightly surreal exploration of quests and history. (I believe this and the above are both technically novelettes, but you can read them in one sitting. Short story ish.) There was also this one amazingly creepy story that I got in a textbook for short story writing, the name and author of which I can't remember; it was done entirely in the second person as someone briefing a new employee on all their coworkers, with all sorts of unsettling details about their personal lives, in friendly business speak. Wish I could remember what that one was. And, heck, just about any short story from Octavia Butler's Bloodchild, though I found "Speech Sounds" most lingering in how unnerving it is--"Bloodchild" itself is damn creepy, but a lot of its body horror comes from male bodies being treated by aliens the way female bodies are treated by everyone--and "Near of Kin" most memorable overall for how oddly touching and sweet it was. (Which especially isn't something I expect from Butler. Ever.)
That's probably it! ...I mean, there's unlikely to be two short stories by that description floating around in a way that two folks on a forum would know about separate ones.
Happened to get it free from Amazon a while ago for my Kindle. It's still free for those Kindle users that want to read it: link.
You can actually find "Hole to China" online over at Lightspeed, still; I just am repeatedly baffled at how to make links with this forum's interface.
Oh, I can't remember the names of half of them. I'll have to open actual books to jog my memory. There's one I remember: Murder Mysteries by Neil Gaiman.
I expect most Lovecraft fans'll have read The Picture in the House but it might be my favourite of his. Mark Twain's The War Prayer is, per wikipedia, maybe so short that one'd call it a prose poem, but hey, that's a good format too.
I dig that story so much that it's not even funny. MY BODY IS READY GAIMAN By that same author, I adore Snow, Glass, Apples.
"Aye, and Gomorrah..." by Samuel R. Delany is my favorite SF story, closely followed by "The Doors of His Face, The Lamps of His Mouth" by Roger Zelazny and "There Will Come Soft Rains" by Ray Bradbury. "Night They Missed the Horror Show" by Joe Lansdale is my favorite horror story. As for literature with a Big L, I really like "Your Lover Just Called" by John Updike, "Big Two-Hearted River" by Hemingway, and pretty much any story randomly picked from Charles Bukowski's Notes of a Dirty Old Man.
Wait, how could I forget? "Even the Queen," by Connie Willis. A fun and funny story about a minor family squabble, which is simultaneously about a massively culture-changing technological innovation that I have not seen explicitly addressed anywhere else. (Implied by other advances in various scifi settings, yes. Explicit? No.) It's sort of my highlight of what science fiction in particular can be, when it stops humping laser guns for a while.
