You read my mind ianzin…
I’m not as fond of hard SF, so I can’t speak to Exapno’s recommendations. My tastes run more toward the so-called speculative fiction that gets lumped in with the “real” SF. That said, your library should have anthologies of each decade’s Hugo and/or Nebula award winners for short stories. That might be a good place to start.
Otherwise, I would recommend Dangerous Visions for a great sample of the Big Names in SF.
Good luck! Once you find out what kind of SF you like, let us know so we can give you better info.
This probably isn’t on anyones list of “best SF short stories EVER,” but it is really good, and it won a Hugo.
“Eurema’s Dam,” by R. A. Lafferty
It can be found in one of the many anthologies of Hugo-winning science fiction edited by Isaac Asimov (Vol. III, as I learn upon searching, published 1977, not apparently out of print. 
)
I nean now apparently out of print.
Personal Favorite: Flowers for Algernon
The short story, not the novel.
I think it’s a crying shame that most people’s experience with it is with the novel.
“Flowers for Algernon” was the first thing which popped into my mind upon readin the title.
Anything from Sturgeon’s anthology A Saucer of Loneliness, most especially the title story, would be in my top 50. His “Baby is Three” wins my award for “most creative use of telepathy” when, forty or fifty pages later, you realize Lone wasn’t making polite chatter.
“Aye, and Gomorrah,” by Delany, if for no other reason than the title.
By the way, the full text of Light of Other Days by Bob Shaw, which truly is a fantastic story, can be found here.
The story that first came to my mind was The Cold Equations.
The only other title I’ve seen mentioned by others that I know I recognize is Flowers for Algernon. I wasn’t aware there was a novel(la) version. I got in trouble for reading the short story in English class because I had finished the assigned reading and was flipping through the text book for something else to read. I don’t recall viewing it as science fiction, but I did like it.
To join in the Sci-Fi eText linkfest:
The Metamorphosis of Prime Intellect
That’s probably because you were taught that if it’s good, it’s not science fiction.
Science fiction was about spaceships, and creatures from outer space, and monorails and flying cars.
So, same to SusanStoHelit. Speculative fiction is as much part of the overall field of science fiction as “real” sf, whatever that is. If you want some good examples of it, let me refer you to my list of stories. Each and every one is speculative fiction.
And robots. Don’t forget the robots! :rolleyes:
I was actually responding to this:
Not your list. The list is indeed spec.fic, and damn fine spec.fic. I had no idea that Kate Wilhelm did short stories too, and I’m going to go find them asap. It was a sad day for sf when she decided to switch to writing Mystery.
I have a number of favorite short stories, but I hesitate to name them “best.” I can’t claim to have read all the stories out there, and even when I am familiar with the stories, I find that I often enjoy the frivolous stories more than the serious ones. With that caveat a far from inclusive list of stories I thought of when I saw this thread title:
“The Cold Equations” by Tom Godwin
“Flowers for Algernon” the original short piece…
“The Vault” by H.P. Lovecraft, not one of his more familiar works - no Cthulhu, just simple horror, and almost plain prose.
“Ender’s Game” by Orson Scott Card, the original short story
“Enemy Mine” by Barry B. Longyear
While Arthur C. Clarke’s “The Star” is a good story, I also love a couple of his other stories more. First the one about the galactic Federation coming to rescue what inhabitants they can from a planet orbiting a star about to go nova - I can’t recall the title to save my life. I think it was his first published story, though. (My mind, to quote the Professor: “like a lumber room - things needed always getting lost.” 
) The other story of his that still gives me a chill is “The Nine Billion Names of God.”
And SusanSoHelit, one place to start would be with one of the recent collections of Nebula or Hugo award winners. In the mid-70’s there was a collection of three books (IIRC World’s Best Science Fiction, Volume Ia, Volume Ib, and Volume II) which had many of the stories listed in this thread. (Including, but not limited to “Arena,” “Flowers for Algernon,” “To Serve Man,” “The Cold Equations,” and many, many other seminal works in the genre. A word of warning - the earlier stories will show the pulp influence on SF - often poor characterization and stilted language, but still an enojyable read. I can’t find the exact collection I’m thinking of, at the moment on Amazon, but my father has it, so if you want the exact data, I can check next time I go shovel their driveway/walk.
For more modern writing, with a focus on ‘new’ science you can also pick up the annual collections of stories nominated for the Nebula Awards.
Are you talking to me or ianzin, who origionally posted:
Good advice, though. Listen to her (it is her, right Otaku?), ianzin.
I’m not entirely sure if/how it fits the OPs criteria, but I can’t keep myself from recommending A Roll of the Dice by Catherine Asaro (link to the story). It ran in Analhabog years ago, I read it in there and was smitten with it, for some inexplicable reason. I think the very end poses interesting questions about the nature of humanity and human habit.
And I’ve completely lost my ability to type. I read it in Analog, not Analhabog, whatever the hell that is…
I think I should have addressed ianzin. I just got confused. I do that easily. :smack:
Thank you, but I do want to correct your assumption - I’m male, not female. I’m not offended, just wouldn’t want to leave you with an incorrect impression.
:o shit :smack:
sorry. Damn gender-neutral names
Goddamn you kids, trying to read in class. Of all the nerve…
I, too, will have to add my vote to the accumulating totals for “The Last Question” and “The Cold Equations”.
Other nominees:
Isaac Asimov’s “Not Final”
Joanna Russ’s “When It Changed”
Murray Leinster’s “First Contact”
Ray Bradbury’s “Dark they Were, and Golden-Eyed”
“Seven American Nights” by Gene Wolfe
haven’t read many science fiction shorts…
A number of favorites have been mentioned here, but two that haven’t is “Nightwings” by Robert Silverberg and “The Moon Moth” by Jack Vance.