Many people focus on dog bites as an example of how pets can be dangerous to others, but cats aren't exactly angels, either. If you have been bitten or scratched by a cat, you need to be on the lookout for infection, not just by run-of-the-mill pathogens, but some cat-specific ones as well. And if that happens, you may have a personal injury case on your hands.
One of the first things that would come under scrutiny is whether or not your interaction with the cat had a reasonable basis. For example, if the cat's owner told you the cat didn't like being petted but you insisted on trying to pet it only to end up with a deep scratch, you can't really claim the incident leading to the scratch was not your fault. After all, the owner tried to warn you.
The same would go for you pestering a cat that was trying to avoid you. Let's say you know a cat is normally OK with you petting it, but now it's trying to avoid you because it's tired or for some other reason. If you continue to pester the cat, it's kind of understandable that it might scratch you because that's how it can make you go away. That's not the cat's fault or the owner's fault.
On the other hand, if you made no move toward the cat, but the cat came up to you and swiped its claws at your arm, leaving scratches, that's different. The cat's owner may be liable in that case, though your lawyer and the court will want to investigate further. (It's possible the cat's owner had done what he or she could to isolate the cat, such as putting the cat in a separate room, only for the cat to escape, for example.)
Also of interest is the severity of the problem. A little scratch that heals quickly and that doesn't get infected is likely not cause for a lawsuit. But a severely infected sratch is an entirely different matter.
Once you've been scratched, you face two risks. One is an infection that can occur from not taking care of the wound properly. But the other is due to a pathogen called Bartonella henselae. This can cause cat scratch disease (aka cat scratch fever). You would not have that disease if you hadn't been scratched or bitten by the cat, and that may mean that the cat's owner would have to pay for medical care and other costs for you.
It's always best to speak to a lawyer before assuming you'd win or lose a case. Don't treat a cat bite or scratch as something that you can't do anything about. You may have legal recourse that allows you to get compensation for all costs associated with treating the wound. Contact a local personal injury law firm for more information and help with your case.