You should know that nowadays "Fräulein" (Miss) is not used any more, not even for unmarried women! It would be offensive to write that, unless you were an old person, or perhaps in an informal letter for fun.
You're right, in writing you can abbreviate Frau to Fr. (and Herr or Herrn to Hr.) but it depends *where* you want to do this. If you are writing a formal letter, you start with your address, and then, a few lines below, you put the woman's, so you begin with "Frau Xy" - it's just not necessary to abbreviate! After the subject you begin with "Sehr geehrte Frau Xy,".
So in a formal letter you don't have to write "Fr.", but it's allowed to use it in other contexts, for example on lists or sth., you just use "Fr." and "Hr."!