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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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You may be referring to the congruence subgroup problem: front.math.ucdavis.edu/0901.4663 –  Ian Agol Apr 18 '12 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.

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