
I have saved you time on research through the Interwebs several times now, and I shall do so again. I need only make use of a magic phrase regularly beats all the search engines the computer savants can throw at you.
“Nobody really knows.”
This time around, I was inspired by making my bed. I will not take time to explain my domestic arrangements, but I make my bed only when forced to by circumstances, as the process involves taking everything apart and then putting the springs where they belong, squaring the mattress on top, and then locating a mattress pad and sheets which will stretch to all the corners and stay in place for more than 24 hours. I have a standard mattress and standard sheets, and yet somehow, the corners will never…let’s get back to the question at hand
I was struggling once again to get the elastic over the corner of the mattress and wondered “Who invented the fitted sheet?” I knew this was a recent innovation, since I was instructed in making beds both by my mother, who eventually worked in a hospital and learned to make a mitered corner for sheets, AND my father, who learned in the Air Force to make mitered corners for sheets. Each, having learned this skill in a fairly demanding environment, was rather proud of this skill, which is one of many I never mastered. I DID accomplish a mitered corner once, but that was probably the last time I even saw a non-fitted sheet.

There are histories of fitted sheets around the Interwebs. The most detailed actually names the lady who in 1957 came up with the concept, but insisted on elastic garters you attached to the sheets before bedmaking. This did not catch on, but later another lady came up with built-in elastic corners, and the world was changed.
Unfortunately, the world of the wide web abounds in nostalgia websites reprinting old advertisements, and there are plenty of catalog listings and newspaper ads for fitted sheets in the early Fifties and even the late Forties. One brochure from 1954 says that fitted sheets have been on the market for five years, which confirms my belief that a LOT of the culture I grew up with was invented in the Post-War World. I have been able to learn nothing about what innovater sewed the top sheet to the fitted bottom sheet, but this does not seem to have caught on.
The first fitted sheets seem to have been for cribs, whose occupants could be counted on to kick and roll and dislodge sheets, which, even leaving out the possibility of diaper leakage, meant a lot of repeat bedmaking. In any case, the Overworked Housewife so often touted in ads for household products (always a winner, since every homemaker feels the pressure of work to be done) was the focus of these campaigns, and the time save by not having to miter the corners day after day were held up as a major selling point.
As more than one humorist has pointed out since, all the time saved in this is then lost by the amount of time it takes to FOLD fitted sheets.

Now, this is a topic on which everyone agrees: there’s a trick to it. After that, they diverge, some experts going in for amazing linen origami. But it boils down to this: you bring the corners of the sheet together in an orderly fashion so that all four are folded together into one little pocket. You now have a MOSTLY flat sheet which you can fold in whatever way appeals to you. SOME experts feel you should then wrap the sheet and its attendant top and pillowcases in another matching piece of fabric which can be used to sort your sheets by color so you can match seasonal or other decorating demands when making the bed. (Guess who uses that method.) Other people suggest you just wad the sheet up into a ball and shove it in a corner of the closet. (No points for guessing who uses THAT one.) Or you could always buy a sleeping bag.