We study the set vol (M, G) of volumes of all representations ρ: π1M →G, where M is a closed oriented 3-manifold and G is either Iso+H3 or IsoeSL^2(R). By various methods, including relations between the volume of representations and the Chern–Simons invariants of flat connections, and recent results of surfaces in 3-manifolds, we prove that any 3-manifold M with positive Gromov simplicial volume has a finite cover Mf with vol(M, f Iso+H3) 6= {0}, and that any non-geometric 3-manifold M containing at least one Seifert piece has a finite cover Mf with vol(M, f IsoeSL^2(R)) 6= {0}. We also find 3-manifolds M with positive simplicial volume but vol(M,Iso+H3) = {0}, and non-trivial graph manifolds M with vol(M,IsoeSL^2(R)) = {0}, proving that it is in general necessary to pass to some finite covering to guarantee that vol(M, G) 6= {0}. Besides we determine vol (M, G) when M supports the Seifert geometry