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.
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3$\begingroup$ You may be referring to the congruence subgroup problem: front.math.ucdavis.edu/0901.4663 $\endgroup$– Ian AgolCommented Apr 18, 2012 at 17:19
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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.
answered Apr 18, 2012 at 21:01