I have got a question that is perhaps not precise in a mathematical sense. Is there a classification of all coverings of the moduli space of Riemann surfaces which are moduli spaces themselves, that is, they parametrize some geometric structure on a surface.

I doubt there is a "classification", but there are some interesting examples. Two which come to mind: Harer's description of the moduli space of a Riemann surface with spin structure; and Torelli space. EDIT: Oops, I forgot to read your title, I just read the text. Torelli space is an infinite rank covering of moduli space. 

