The way in which domestic cats are kept and bred has changed dramatically over the last two centuries. Notably, a significant number of cats are kept indoors, most of them are neutered and many are selectively bred. This likely has consequences for their welfare. A few studies link housing, neuter status and breeding in cats to risks of welfare problems. However, the study presented here is the first to quantify the risks and document the prevalence of risk factors. It builds on results from a questionnaire sent to a representative sample of the Danish population. Using the responses from cat owners who keep cats in the home (n = 378), the paper aims to investigate how indoor confinement, neutering and selective breeding affect health, behaviour and other factors relating to cat welfare. The paper reports that confined cats had significantly more behavioural problems than free-roaming cats; that a smaller proportion of the free-roaming cats suffered from the behavioural problems investigated; and that entire cats had significantly more behavioural problems than neutered cats. Finally, significantly more purebred cats than domestic shorthair cats were found to have diseases. Being confined, being intact and being purebred are therefore significant risk factors for behavioural or health problems associated with reduced welfare in privately owned cats.