Properties of hash functions

Does anyone know the following, or can someone point me to the proper references? What properties are preserved by hash functions? For example, take x = MD5(Y). Now, say that Y is a quadradic residue mod N, is it the case that x is also a quadradic residue mod N? Are there any interesting properties that are preserved by frequently used hash functions? Thanks a lot for any help, Avi Rubin