I’m new here, but I’ve been lurking long enough to respect the literary breadth of the Dopers. I’m a high school English teacher in a small school, and our literature books are old and boring. With the state in debt, there is no money for new texts, but I’m trying to put together some short stories that are thought-provoking, important, classic…
What are your suggestions? Stories that are in the public domain would be great, as I could print them out.
“Nightfall” - Isaac Asimov
“There Will Come Soft Rains” - Ray Bradbury
The Cold Equations" - Tom Godwin
That’s definitely one on the list. And Lovecraft’s Dagon.
I always liked Guy de Maupassant’s The Necklace and O Henry’s The Ransom of Red Chief.
Edited to add, Asimov’s The Last Question, but it’s not public domain.
The Most Dangerous Game
It’s a classic for a reason.
Cathedral, or many others by Raymond Carver.
*The Diamond As Big As The Ritz
- by F. Scott Fitzgerald
Edgar Allen Poe’s The Tell-Tale Heart
“The Curious Case of Benjamin Button” by F. Scott Fitzgerald.
“A Logic Named Joe” by Murray Leinster.
Love is a Fallacy by Max Shulman.
No, it’s not exactly a “classic” and it’s probably not “important” at all, but it’s a great, fun read, and it’s wonderfully educational, using easily understood examples of classic logical fallacies. The narrator has a strong voice, and the writer makes some interesting word choices and sentence structures, which I think makes it interesting from a lit standpoint, and I’m all for increasing the critical thinking and logic exposure to high schoolers, even if they won’t formally study these things until college.
And it’s free.
PD stories:
“The Lady or the Tiger” by Frank Stockton. Perfect for class discussion, too.
Anything by Poe. “The Purloined Letter” might be good
“The Birth-Mark” by Hawthorne
“The Problem of Cell 13” by Jacques Futrelle.
“The Hands of Mr. Ottermole” by Thomas Burke – one of the greatest mystery short stories ever written (though it’s still be under copyright).
Stephen King, “The Last Rung on the Ladder.” It’s more on the thought-provoking end than classic or important, but it includes some technical points worth discussing (characterization and such) and will also, with any luck, be appropriately unsettling. 
Cynthia Ozick - The Shawl
John Cheever - The Swimmer
Washington Irving - Rip Van Winkle
Tim O’Brien - The Things They Carried
Franz Kafka - The Metamorphosis (not a short story, but short fiction)
Joyce Carol Oates - Where are you going, where have you been
DH Lawrence - The Rocking Horse Winner
Ernest Hemingway “The Short Happy Life of Francis Macomber”
First that comes to mind is Poe’s “The Cask of Amontillado”; and “After Forty Years,” by (I believe) O. Henry.
I’d also recommend looking into Stephen Leacock. “My Financial Career” is fun, but he wrote a number of other stories that would be suitable. You may also wish to try a chapter or two from his Sunshine Sketches of a Little Town–this is generally regarded as a novel, but while the same characters appear in each chapter, each chapter can be regarded as a separate story.
The nice thing about Leacock is that his stuff was written in the early 20th century, and is now in the public domain. You can read and download the text for free at the Gutenberg Project.
Harrison Bergeron by Kurt Vonnegut is a favorite of mine…
Also by Asimov: “The Fun They Had”
Also by Bradbury: “All Summer in a Day” and “The Veldt”.
All of these actually appeared in my reading textbooks as a kid. Read “The Veldt” in junior high, but the other two were in elementary school readers.
flowers for algernon
The Secret Life of Walter Mitty", by James Thurber. Loads of critical analysis and study guides online, as well.
“A Rose for Emily” by William Faulkner…this was used in my AP English class.