There are so many excellent stories out there, I shall have to give this some thought.
Here are a few stories that spring to mind, these are all good, worth looking out for.
"Four In One" by Damon Knight, a really brilliant story which starts with the hero being eaten alive.
"The Marching Morons" & "The Little Black Bag" both by C.M.Kornbluth, both set in the same universe, when I look at some of modern day TV I can see "Morons" coming true.
"The Wordprocesser Of The Gods" by Stephen King, a man is given one tinkered together by his late nephew, which he then finds can alter reality.
"I Made You" by Walter M. Miller Jr, a man is trapped on the Moon, being hunted by a robot tank, interestingly it's written from the tanks point of view.
"Little Lost Robot" by Isaac Asimov, what can happen when you mess with the three laws.
"The Liberation Of Earth" by William Tenn, a very heavy satire about little people caught up in a big fight.
"Big Joe And The Nth Generation" also by Miller, set on a terraformed Mars where civilization has lost all the old knowledge.
"All Mimsy Were The Borogoves", "The Proud Robot", "Time Locker", "The Townky", these are just a few of the outstanding stories of Henry Kuttner & possible C.L.Moore, they were husband and wife and would some times collaborate on stories but publish them under Henrys name.
"Window" by Bob Lehman (think I've got the name right), this is SF with a very strong Horror element, very creepy and memorable.
These are just a few for now, I'll try to think of some more.
It's nice to see a thread devoted to short stories, they seem to get ignored a lot of the time in favour of novels.
But to my way of thinking a good s/s is as memorable as any epic novel.
And they stay in the memory just as long.