We study two-dimensional nonlinear sigma models in which the target spaces are the coset supermanifolds U(n+m|n)/[ U(1)× U(n+m-1|n)]≅ CP n+m-1|n (projective superspaces) and OSp(2 n+ m|2 n)/OSp(2 n+ m-1|2 n)≅ S2 n+ m-1|2 n (superspheres), n, m integers, -2⩽ m⩽2; these quantum field theories live in Hilbert spaces with indefinite inner products. These theories possess non-trivial conformally-invariant renormalization-group fixed points, or in some cases, lines of fixed points. Some of the conformal fixed-point theories can also be obtained within Landau-Ginzburg theories. We obtain the complete spectra (with multiplicities) of exact conformal weights of states (or corresponding local operators) in the isolated fixed-point conformal field theories, and at one special point on each of the lines of fixed points. Although the conformal weights are rational, the conformal field theories are not, and (with one exception) do not contain the affine versions of their superalgebras in their chiral algebras. The method involves lattice models that represent the strong-coupling region, which can be mapped to loop models, and then to a Coulomb gas with modified boundary conditions. The results apply to percolation, dilute and dense polymers, and other statistical mechanics models, and also to the spin quantum Hall transition in non-interacting fermions with quenched disorder.