According to the American Society for the Prevention of Cruelty to Animals (ASPCA), while neutering a male can resolve territorial marking (spraying/urination & defecation), it doesn’t always eliminate it completely. Females have also been known to ‘spray’ (and/or defecate), neutered or not; however, this is far less common.
http://www.aspca.org/site/PageServer?pagename=pets_urinespraying
If your cat sprays/urinates or defecates inappropriately, even if he/she isn’t showing outward signs of illness or infection, it’s strongly recommended that a vet visit be scheduled urgently. When urinary tract (or intestinal) problems present, there may be very little time to treat it successfully. However, when caught early, there’s a much higher chance of recovery.
In an extensive publication (by David Senior, LA University) encompassing two (multi decade) studies, it was found that Feline Lower Urinary Tract Disease would cause cats to urinate (and defecate) in inappropriate places; however, without other symptoms such as urgency, pain upon urination (often shown by meowing deeply/loudly while urinating), blood in the urine, refusal to eat/drink, a vet visit is still warranted to be certain there’s no early infection or blockage, but behavioral motivations would likely be the cause.
Karen Overall, MA, VMD, PhD, Diplomate ACVB, ABS Certified Applied Animal Behaviorist (among other titles) extensively researched, studied and published her findings that such behavior may be an aggressive acting out. Dr. Overall insists upon a full work up for overall & urinary health with regular follow-ups while behavior modification is undertaken.
It’s just prudent to keep making sure there’s no underlying disease or infection that may have been missed (false negatives, etc) upon initial exam.
Once it's decided that this is not because of illness or disease, start exploring why he'd want to 'go' in the areas he chooses. It could be that he likes the feel of the rug (or whatever the material) as his litter.
Could he be smelling another animal? Is the area or items he chooses from his favorite human (or least favorite ?). He may be marking his 'ownership' of that place or person just like he could be marking his 'dominance' over it or them.
To help curb it, remove any trace of his previous marking and remember, just because you can't smell it, doesn't mean he can't! He's got about 200 MILLION scent receptors to our little 5 million Images available only to Customers.
Once you're relatively sure the areas are clean, for a while (just to break the habit) lay down an old (clean) shower curtain or even some plastic tarp. It might not look good, but then, neither does his using it like a litter box right?
Make sure his litter box is kept meticulously clean. He may be choosing other spots because it's too 'crowded' in there! Cats are known to be very picky this way (we had one who refused to use the box if even ONE clump was in it; fortunately, he used the shower stall as an alternative, which was pretty easy to clean up).
With this information and options, you should get control of the whole thing in a relatively short period of time and get back to a nice, harmonious and stink-free environment Images available only to Customers.
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Many have forgotten this truth, but you must not forget it.
You remain responsible, forever, for what you have tamed.