Is there in the literature a list of three dimensional spherical, good orbifolds covered by nil, Sol and E3, and their algebraic topological invariants? (Homology, orbifold fundamental group). 



Enumeration of three dimensional spherical good orbifolds covered by Nil, sol and E3
Yes, this was done by Bill Dunbar (for Euclidean orbifolds in his Princeton thesis, the general result is in an [impossible to find 1988 paper](https://www.dropbox.com/s/o2p056132rvl6qh/dunbar.pdf?dl=0):


*Dunbar, William D.*, [**Geometric orbifolds**](https://eudml.org/doc/43193), Rev. Mat. Univ. Complutense Madr. 1, No.1-3, 67-99 (1988). [ZBL0655.57008](https://zbmath.org/?q=an:0655.57008).


