Tattiness is the noun for the state of being tatty — the shabbiness of a thing, the fact that it is worn out, faded, and not cared for. You use it when you want to talk about that quality itself rather than describe an object: 'the general tattiness of the building', 'I was embarrassed by the tattiness of my old suitcase'. It is often used about places — a hotel, a station, a shop — where things have simply been let go over the years. It is uncountable, so no 'a' and no plural. Like 'tatty', it is British and rather informal, and it usually carries a faint note of complaint or apology.