All by Stephen King, in my opinion the master:
The Long Walk (Bachman, a novella)
The Mangler
Mrs. Todd’s Shortcut
The Mist
The Lawnmower Man
The Langoliers
Rage (Bachman, a novella)
Quitter’s Inc.
OK, that’s more than 5 and Rage and The Long Walk were actually novels but they were short ones. 
I’ll go with your listing and add Apt Pupil
A couple of my favorites have already been named.
I’ll go with:
Sredni Vashtar by Saki
Thrawn Janet by Robert Louis Stevenson
There Will Come Soft Rains by Ray Bradbury
The Rocking Horse Winner by D. H. Lawrence
The Necklace by Guy de Maupassant
The Little Movement, by Philip K. Dick
Goldfish Bowl, by Robert Heinlein
Ginny Sweethips’ Flying Circus, by Neal Barrett, Jr.
The Three Horsemen of the Apocalypse, by G. K. Chesterton
The Awful Reason of the Vicar’s Visit, by G. K. Chesterton, which also boasts the best title ever.
Here’s a sample from the last one.
Now go read the rest.
The Mangler: Stephen King
Quitters, Inc: Stephen King
Saki: Can’t remember the title, but the last line is Wolves!
The Most Dangerous Game: Richard Connell
Appointment in Samarra: Somerset Maugham version, if that counts
I don’t read much short fiction these days, but over the years my favorites have been:
Sherwood Anderson, “Sophistication”
John Barth, “Lost in the Funhouse”
Jorge Luis Borges, “Tlön, Uqbar, Orbis Tertius”
James Joyce, “The Dead”
John Updike, “Pigeon Feathers”
The Interlopers
What does this mean? AIS is a novel by John O’Hara–the title is taken from a line of one of Maugham’s plays, “Sheppey.”
“I Have no Mouth and I must Scream” – Harlan Ellison
“A Good Man is Hard to Find” – Flannery O’Connor
“Lost in the Funhouse” – John Barth
“Mother Hitton’s Littul Kittons” – Cordwainer Smith
“And I Awake to Find Me Here on the Cold Hill Side” – James Tiptree, Jr.
No particular order and not checking to see how (I’m sure) common they are in responses:
-Occurrence at Owl Creek Bridge (Ambrose Bierce)
-The Life You Save May Be Your Own (Flannery O’Connor)
-A Christmas Memory (Truman Capote)
-The Library at Babel (Jorges Luis Borges)
-Shingles for the Lord (William Faulkner)
I love “Sophistication,” but I do not see how it can be called a short story.
http://www.k-state.edu/english/baker/english320/Maugham-AS.htm
It isn’t even a short story- more like a long paragraph. I have never read the O’Hara novel.
Ah yes, so it is The Interlopers, thanks, Le Ministre.
I love these kinds of threads. They make me think and they make me look through my books. Can I list six? Several have already been named:
A Christmas Memory by Truman Capote (from, among other collections, The Complete Stories of Truman Capote)
Brokeback Mountain by Annie Proulx (from the collection Close Range: Wyoming Stories)
The Third and Final Continent by Jhumpa Lahiri (from the collection Interpreter of Maladies)
A Vermont Tale by Mark Helprin (from the collection Ellis Island and Other Stories)
The Things They Carried by Tim O’Brien (from the collection of the same name)
The Witness For The Prosecution by Agatha Christie (from the collection of the same name)
Several ones that I might have said have already been listed, so here’s my five that haven’t been mentioned (since my Top Stories List is much longer than five):
What We Talk About When We Talk About Love, Raymond Carver
The Demoness, Tanith Lee
Deathbird Story/Ahbhu, Harlan Ellison (cannot read it without crying)
Rape Fantasies, Margaret Atwood
and all-time my favorite, The Horse Dealer’s Daughter, D.H. Lawrence
I’m also heartened to find several of the stories I teach to my middle school students have already made people’s lists.
“The Scarlatti Tilt” by Richard Brautigan
“The Shadow of the Vulture” by Robert E. Howard (I’m a lowbrow, I know)
“Kneel Down and Lick My Feet,” Amy Yamada
“The Whore of Mensa,” Woody Allen
“Gianni,” Robert Silverberg
Wickedness (the blizzard story) from Nebraska by Ron Hansen
The Paperhanger by William Gay
My Dead Dog Bobby by Joe Lansdale
The Crowd by Ray Bradbury
Stick Woman by Edward Lee
The Reach by Stephen King
“Harrison Bergeron” by Kurt Vonnegut
“A Sound of Thunder” by Ray Bradbury
“The Fall of the House of Usher”, “The Tell-tale Heart” and “The Cask of Amontillado” by E.A. Poe
Just about every Conan-based story by Robert E. Howard, especially Red Nails and The Thing in the Crypt.
Both immediately came to my mind when I saw the thread title. The Benet was good, but the Maughm was fantastic. May I add:
Shakespeare’s Planet by Ray Bradbury (I’m pretty sure it was Bradbury)
**The Cisco Kid **by O’Henry (nothing like the kinder/gentler Cisco Kid that Hollywood gave us)
**The Man Who Corrupted HadleyBurg **by Mark Twain (Jeez, Twain understands [understood] humanity)
Alibi Ike by Ring Lardner (many have forgotten just how great Lardner was with a short story)
Rain by the aforementioned Maughm (possibly not as touching as “Mr. Know-All” but very powerful)
Richard Matheson, “Dress of White Silk”
Brian Aldiss, “Super-Toys Last All Summer Long”
Shirley Jackson, “The Lottery”
E.A. Poe, “The Tell-Tale Heart”
William Faulkner, “A Rose for Emily”
Old classics, some new. 
“Time Considered As A Helix of Semi-Precious Stones” --Samuel R. Delaney
“The Man Who Travelled in Elephants” --Robert Heinlein
“The Short Happy Life of Frances Macomber” -Ernest Hemingway
“The Secret Life of Walter Mitty”–James Thurber
“The Green Hills of Earth”–Robert Heinlein. Still makes me cry. Every time.