How many different types of nasal cancers are there?

Dozens: There are >50 types of sinonasal tract tumors, many of which are malignant. Cancer is the generic term for a malignant tumor, but they can be divided into carcinoma, lymphoma, sarcoma, melanoma, among others. Squamous cell carcinoma is by far the most common malignant type. It is usually managed by surgery and radiation. There are several grades of tumor along with stages.

More than one: Cancer can arise from different types of cells which is what gives it its name. Skin cancer on the skin surface is different than the cancer arising from the mucus membrane of the nose, which further has glandular cells and surface cells and pigmented cells in it. So a nasal cancer can be squamous type, adenoid cystic(glandular type) or a mucosal melanoma, each with a different behavior and course.