Short stories don't require as much worldbuilding as novels. The world doesn't have to be as large or as complete because the reader's only going to be seeing a portion of it. I start with a few ideas, usually consisting of a character, a basic view of setting and a loose direction for the plot. Often, I don't even know how or when it's going to end.
Most of the worldbuilding details come in after I've written the first draft. I add them in and shape them during editing. So, I go through a lot of edits, filling in details and revising the worldbuilding until it's streamlined to the story. Even if it's not visible to the reader, all the pieces fit together and relate back to theme and plot. Or, at least, as many of the pieces as I can get a hold on.
I build worlds around my short stories. That's not to say that the worldbuilding serves the plot, however, because the plot may change as well. The two feed off one another.
I usually don't do much research either. If I already know something's going to be important to the story, I'll do some general research on the topic. Otherwise, I wait until I have that first draft to draw from. I just never know what's going to be important to the story until I've written it.
I'm working on Fated right now--literally around writing this--and I never knew sledding was going to be at all important. Now that I've written that section, I know that there are some things I'll need to research before I edit this draft.
I think research and worldbuilding also go hand in hand for me. As I research something, sledding for example, I learn more about it and in turn more about the world. What kind of terrain is best to sled on? What kind is most challenging? Do I want their journey to be easy or challenging? Once I decide that (challenging, of course!) I have a better idea of the terrain and I can describe it in more detail.
If I built the world beforehand, I'd have tons of unnecessary details and no space to cram them in. Basically, I worldbuild short stories after the first draft because otherwise I'd end up writing a novel. ;)
How about you guys? How much of a world do you start with, before you start writing a short story?