The problem is probably easily resolved and comes down to a short list:
Type of box
Type of litter
Number of boxes
Cleanliness of the box
You may be one of the lucky few that has a cat that isn’t choosy but nearly 10% of cat surrenders are due to litter box issues and some cats are abused because of their elimination choices.
Think of the litter box as a toilet. Do you care about the comfort of the seat, the height of the seat, how easily it flushes, how many people use the same toilet, and how it smells? Of course you do. Your cat has an even greater desire for cleanliness and a much keener sense of smell. He has needs.
Many people make the mistake of locating the litter box in the laundry room right next to the noisy washing machine or they put the litter box in the cramped corner of a closet or in a tight spot between the commode and the tub. Their cat must be a real trooper to stick with a box that is so poorly located for him. Please don’t put the box anywhere near his food and water. You wouldn’t like to eat in the bathroom either.
Often a litter box is too small for the cat. He must contort himself into awkward positions to eliminate within the box. Please get a big box for him. The box should be large enough for him to turn around in without hanging over any side.
The litter may be rough on his feet, or allergenic, or dusty. Many varieties of litter are made from pelleted pine, wheat, or corn that can be allergenic to some cats. Perfumes can be allergenic or upsetting to your cat. Clay is more inert but some kinds are quite dusty. If you aren’t sure what your cat prefers, try offering different types in different boxes and see which he uses. A cat may temporarily be put off his litter box after being declawed because of some pain.
The litter should be deep enough that urine does not reach the bottom so, if your cat likes to dig deep, you need deeper litter. This is why scoopable litter is often a better choice than non-scoopable. The urine is absorbed before it pools on the bottom of the box. Typically 2 - 4 inches is good.
Rule of thumb is one more litter box than you have cats and, if possible, a litter box on every floor of the house. You wouldn’t want to have to dash long distances to use the toilet. And cats can be bullies and may try to block the use of a litter box by a more timid cat. When given the choice between pooping on the towels and having a fight, most cats will pick the towels.
Lastly, but probably most importantly, the box must be scooped EVERY day. Your cat doesn’t like to maneuver around previous deposits and his sense of smell will deter him from even entering a dirty box. Much is done to try to help people mask the smell of the box, including hoods, vents, filters, perfumes, etc. but, in reality, they don't do much to help your cat. And, the box should be CLEANED once a week. This means that you dump the litter out and wash the box with a very mild soap and hot water. Never use ammonia. Ammonia is what gives urine its offensive smell. Your cat will notice.
An important caveat - If your cat stops using his litter box suddenly or seems to be trying to use the box with frustration, he may have a serious medical issue like a bladder infection, a blockage, diabetes, or kidney failure and should be seen by a veterinarian immediately.