All cats are capable of spraying, whether they are male or female, entire or neutered, and, put simply, it is one of their ways of communicating something that they feel strongly about, whether they are trying to get their point across to other cats or make their feelings known to their humans.
Entire male cats are very territorial, and it is completely natural for them to spray vigorously, which is one of the reasons that they do not make good pets, and should only be kept entire in specially-built cat housing by experienced breeders wishing to use them as stud cats. A stud owner will wash their stud house on a daily basis for reasons of hygiene, although a working stud cat will continue to produce a very pungent smell.
Breeding female cats will also spray to attract a male when they are coming into heat, and the combination of the oestrogen in their urine, together with secretions from the anal glands under their tails, leads to a very strong-smelling discharge. Breeders will be used to this and will have their own ways of dealing with it, but if a female is not going to be a breeding queen, early neutering should help to eliminate this potential problem.
Neutered cats will often spray in the garden to mark their territory, and although this behaviour is less common indoors, there is generally a tangible reason for it if it does happen. If you have more than one cat, a natural leader will emerge and so long as your other cats appreciate this and accept that they are not the 'boss' and do not challenge the leader's authority, incidents of spraying will probably be rare. However, the dominant cat, which may be male or female (it tends to relate more to temperament than gender in neutered cats), will probably spray in the initial stages of assuming dominance, in order to confirm the position.
Another cause of spraying from neutered cats can be linked to a significant change in the home, and it is your cat's way of telling you that they are not happy about what is going on. It could be a move of home, a new partner, the arrival of a baby, the addition of a new pet, a long-term visitor, disruptive building work, or even just decorating. Cats like to have a regular routine and are more sensitive to changes in their lifestyle than many people realise. If it appears that a major change has caused your cat to start spraying, try to minimise the effect on them as far as possible - you will probably need to spend more one-to-one time with your pet to reassure them that all is still well, and they are still loved.
Cats sometimes feel threatened by others in the neighbourhood, and if you have a cat flap, your pet may well spray around it to discourage the interlopers from entering their home. But if a neighbouring cat enters your house, particularly when you are out, and threatens or bullies your cat, he or she will probably spray out of fear, and in a vain attempt to discourage the unwanted visitor. If this happens, it might be worth installing a magnetic cat flap, which is activated by a special magnetic collar worn by your pet, although this is not foolproof - some cats just don't like wearing collars or another cat may follow yours in at great speed after a chase, and will then be unable to get out again. In this situation, you might need to think about keeping your cat indoors (with a litter tray) whilst you are not there and then only letting them out when you are available to keep an eye on things.
Some cats are unfortunately highly-strung and nervous, often for no apparent reason, though sometimes because they have been re-homed, were previously feral, or sadly, have even been mistreated. Nervous cats are likely to spray simply because they are frightened, or in an attempt to make their environment completely their own so that they feel a little safer. In these situations, your cat may well settle down in due course and stop spraying, or it may be that plenty of loving attention will help to make them feel more established. However, you need to be aware of your cat's needs, and if he seems to be telling you that he would prefer not to be handled and doesn't relish the additional attention, respect his wishes - he may well come round in due course, and if he doesn't, it is better for his state of mind to simply let him be.
It will probably help if you limit the rooms where your cat is allowed if he is a frequent sprayer, not simply for the cleanliness of your house, but also to help your cat feel more secure within a smaller territory. One of a cat's natural ways of marking its territory (which is common to all cats, whether or not they spray) is by rubbing themselves around furnishings, walls and doors, and even around you, their owner, which releases a natural scent containing pheromones, that send messages about security and familiarity to the cat's brain. You can help this process in the case of a cat that sprays by rubbing a soft dry cloth gently round the cat's face and then rubbing the cloth in the areas that seem to presenting the problem. Or you can get a plug-in diffuser from your Vet that slowly releases synthetic pheromones into the atmosphere (and which are not discernible to the human nose), which will help to relax your cat.
A few neutered male cats spray because of a hormone imbalance. If you think this may be the case for your cat (and you are unable to see any other reasons for them to spray), your Vet might prescribe a low dosage of oestrogen to redress the balance. Don't worry - this will not turn your male neuter into a female, or largely alter their behaviour and temperament, but it might just stop them spraying and may at least be worth a try. Oestrogen can sometimes affect the shape of your cat very slightly, particularly the head shape, and if you have a pedigree male neuter that you show, you may need to take this into account first.
The spray from a neutered cat is nothing like as strong and pungent as that of an entire cat, but nevertheless you should wipe it up as soon as you notice it to prevent a smell in your house, as well as to prevent the cat spraying in the same place again, or attracting germs that could lead to diseases. There are plenty of proprietary cleaners for this purpose available from pet stores and supermarkets, or use a good household cleaner, making sure you don't use anything that is poisonous to cats. You could also try a solution of white vinegar and water (one part vinegar to two parts warm water) or biological washing powder in water. Don't use anything containing ammonia in this situation as feline urine comprises a high level of ammonia and so it will encourage, rather than discourage, your cat to spray.