The one thing worth knowing before you buy a single fitted sheet is your actual mattress depth, not the one you assume. Measure it with a tape, corner to corner, through the mattress itself, because the difference between a twelve inch pocket and a sixteen inch pocket is the difference between a sheet that hugs the corners all night and one that pops off at three in the morning and ends up balled up at the foot of the bed by sunrise.
If you've got none in the house that fit properly, start with a plain cotton percale in a neutral color, white or a pale gray, something that works in a guest room, a kid's room, and your own bedroom without you having to think about it. Percale breathes, it softens with washing instead of pilling, and it doesn't come with the slight sheen that makes cheap sateen look like it's trying too hard. Skip anything that advertises an enormous thread count on the package. Past a certain point that number just means more cheap short fibers crammed in, not more quality, and a sheet built that way will pill in the dryer within a couple months. A well made fitted sheet has deep elastic sewn all the way around the hem, not just at the corners, generous all around elastic in the right spots, and a seam that lies flat under you instead of a ridge you can feel through the mattress pad.
Where it goes matters more than people treat it like it does. A crisp cotton or linen blend fitted sheet suits a bedroom that gets warm at night, the kind with a window unit working overtime in July, because natural fiber pulls heat away from your skin instead of trapping it. Flannel or a brushed cotton earns its spot in a colder bedroom, a converted attic room, anything without great insulation, where you want the bed to feel warm the second you get in rather than ten minutes later. Rental apartments do well with a couple of all around sets in different weights, since you're often stuck with whatever heating or cooling the building decides to give you that week, and swapping the sheet is a lot cheaper than arguing with the landlord about the thermostat.
A house with a guest room that actually gets used, a kid who grows two mattress sizes before anyone notices, and a closet where nobody wants to spend ten minutes matching elastic to mattress depth every Sunday, that's the home a good set of fitted sheets is built for.