You use single-handedly when one person did the whole thing, with no help. She single-handedly organised the concert — she did it all. It is a strong, slightly dramatic word; people reach for it when they want to praise someone (or sometimes to complain) that the work sat on one pair of shoulders. Though actually, for small everyday acts we more often just say by herself or on his own. Save single-handedly for bigger efforts. It is the adverb; single-handed is the adjective.