My cat has scabby spots all over her nose. Her behavior seems normal but her nose looks awful. She is a house cat and never goes outside except on a screened-in front porch.
The first thing I would want to know is whether she is a white cat or has a white nose. If so, I would advise a vet visit. Cats can develop a form of skin cancer on white areas that have been exposed to sunlight (such as sleeping in windows).
If that is not the case, my next thought would be allergies and perhaps she is rubbing her nose because it is itchy.
The alternative is that perhaps she rubs her nose frequently against some object such as the screening on your porch or a favorite sofa. Cats do that sometimes just out of habit. In the case of the screened porch, they may do it because they want to get to other animals they see (squirrels, rabbits, etc.)
The appearance of the nose you describe can develop quickly. Occasionally, a cat that is brought into my hospital will arrive with a traumatized nose from rubbing it against the carrier door.
Thank you for writing,