As a bloke with 3 of my own and most of my adult life spent volunteering with Cats Protection, not much of a deal breaker for me (and for many men that I know with cats) - providing of course looked after properly.
So things like indoor only cats, where it isn't for medical reasons, is a problem for me (fine if they choose not to go out despite having access of course - that's the cat's choice).
Breed cats (unless rescues) are also problematic for me as I find the breeding of animals to specific artificial standards for human pleasure pretty repugnant, and would struggle with anyone who chose to buy one and thereby support the trade.
Strangely, as someone who has been starting to dip their toe into the swampy morass of OLD for the first time ever, I have been somewhat surprised by what I have seen. Very, very few women mention cats in their profiles, but many mention dogs. Given that it's extraordinarily unlikely that only non-cat owned women are finding themselves in their fourties and single, I can only assume that there is a mass assumption that men like dogs and hate cats, so (think Basil Fawlty here, 'Don't mention the cats'... Personally I would think it would be a great filtering mechanism, so please put the cats in your profiles!
Obligatory cat pics inserted... (Wriggly Don cat is the black and white, who has just left me covered in white hair following his morning cuddle).