What type of groups does Microsoft's U-Prove use (Schnorr... etc?)
U-Prove Recommended Parameters describes the groups used by U-Prove.

For the subgroup variant it references Appendix A.1.1.3 of FIPS186-3 which is about groups for finite-field based DSA. AFAIK these groups are Schnorr groups, even though NIST never refers to them as such.

The ECC variant uses standard NIST curves such as P-256, P-384 and P-521.