I don't think there is a clear rule like Naoki-san wrote. But, roughly speaking, ズ（ず） is used much more than ヅ（づ）. There are basically three situations as follows: case#1 ず or づ　comes first: ex)ずいぶん、ずっと、づけどん（づけ丼） case#2 ずor づ comes in the middle as an euphonic change of unvoiced to voiced sound（連濁（れんだく）） ex)すず、すずしい、すずき、つづき、つづら case#3 ず or づ comes at the end: ex)　くず、あいづ（会津） Now that づ mostly exists in case#2. づ in case#1 rarely exists and づ in case#3 mostly exists in names of places, which doesn't happen a lot. (About づ in case#1, I can only think of づけどん, which is pickled raw fish on rice.) I hope this helps you even a little.

I think there is no rule. So you should remember every single spellings. It's really hard, but you'll see some patterns which I cannot explain for some reason... GOOD LUCK :)