Obviously couldn't try every spelling at once, but are borscht, borsch, borshch, borsht, and bortsch all accepted as correct spellings? As far as I know, those are all correct. :)
Delicious, by the way! I had some in Canada awhile back, home cooked by my friend's Polish mother. I imagine there are some regional differences to the recipe, of course, so hopefully I'll get to try some Ukrainian борщ too someday!