Controllability and impulse controllability indices for singular systems are defined via the concept of minimal indices for modules over polynomial rings. These indices are invariant under the transformations of restricted system equivalence and feedback equivalence. They yield a characterization of controllability and impulse controllability of singular systems analogous to the state-space situation and form a complete list of invariants for impulse controllable systems under feedback equivalence. A feedback canonical form described by these invariants is constructed explicitly.