I've linked to the story on-line if I can find it.
1. The Colour Out Of Space by HP Lovecraft
A beautiful story, recounting the impact of an alien infection or radiation on a small farm in rural New England. SF told as Gothic. Well worth a read.
2. The Winter Flies by Fritz Lieber (also known as 'The Inner Circles')
Almost not SF. A tired business man daydreams while his family relax in the evening. Or, possibly, a family is menaced by evil forces. The ambiguity is maintained to the end. The winter flies.
3. The Heat Death of the Universe by Pamela Zoline
Also almost not SF. A woman ruminates on entropy as she cleans the house. Brilliantly written. I wrote a review of it here. and looking for that link I found this lj page about the story.
4. Intracom by Ursula K LeGuin
When I first read this I didn't know there was fanfiction, and I was thrilled to bits. This is Le Guin's Star Trek fanfiction. Well, almost. She demonstrates (and this was back in 1974 or something) what is now commonplace - that the crew members of a space ship are aspects of the personality. As I like to say 'The Hull is the Skull'.
I read this back in the seventies, and thought about it often since. It is only now that I discover it isn't called 'God's Motorway' although that's what it's all about. The climax of the story happens on the M6 just north of Coventry - yay for the West Midlands. Everyone is this story is deluded about the nature of reality, leading to constant anxiety, but when they find out the truth it doesn't help them. Cheery stuff.
6. Magic for Beginners by Kelly Link
Oh, you can't do better than this in modern SF. Fandom as magical practice, and an incredibly well written story too. Has won bags of awards. A must-read.
7. Houston, Houston, do you read? by James Tiptree Junior
A radical feminist SF story, setting the standard for that whole sub-genre, about a world without men, and what happens when modern men (actually men of the early 1970s) encounter it. I often think of this story when I see how men behave to each other, that paralysing hierarchical deference some of them can't seem to snap out of.
8. A Sound of Thunder by Ray Bradbury.
Bradbury is a great SF writer, and this is his most famous short story. Man, it's so famous it was made into a Simpsons episode. This is the one where a guy goes back to the Jurassic, treads on a butterfly, and comes back to the present to find that - oops - he's changed history. I read this when I was almost too young to read.
9. The Cold Equations by Tom Godwin
This is not the type of story I like: it's the type of story beloved by engineers and computer scientists who think they are the alpha class. Nevertheless it is probably the most discussed SF short story ever, and I am very interested in it. Its basic idea (shuttle craft low on fuel - must we eject one of the passengers?) has been frequently borrowed, not least by Blakes 7.
This must be available online but I can't find it - can someone send me a link to it?
10. Inconstant Moon by Larry Niven (link is to partial version only)
I kind of dislike Larry Niven but this is a classic story, where our hero deduces, from the brightness of the moon, that the sun has gone nova and a wave front of boiling air is racing towards the dark side of the Earth. Good premise.
ETA - I have managed to nab two minutes on the computer to say that temeres is right to nominate Harrison Bergeron by Kurt Vonnegut. It is more famous and influential than 'Inconstant Moon' and Vonnegut is more to my taste than Niven, so I'd probably swap. Oh, and I've defined 'short story' liberally rather than restrictively.