I put Single correct? My fiance' seems to think unmarried partner is appropriate. Help?
No, unmarried partner is for a completely different visa, whereby you are both living together legally in the same country and have no intention of getting married.
You are not married yet (you are coming to the UK for the purpose of getting married), so you put single.
And just to make sure you know (and haven't selected the wrong option), at the beginning of the application, where it asks for your purpose of application etc. you select:
Settlement -> Settlement -> Marriage