For a general singular system S-e[E, A, B] with an associated pencil T(S), a complete classification of the right polynomial vector pairs ((x) under bar (s), (u) under bar (s)), connected with the N-r{T(s)} rational vector space, is given according to the proper nonproper property, characterising the relationship of the degrees of those two vectors. An integral part of the classification of right pairs is the development of the notions of canonical and normal minimal bases for N-r{T(s)} and N-r{R(s)} rational vector spaces, where R(s) is the state restriction pencil of S-e[E, A, B]. It is shown that the notions of canonical and normal minimal bases are equivalent; the first notion characterises the pure algebraic aspect of the classification, whereas the second is intimately connected to the real geometry properties and the underlying generation mechanism of the proper and nonproper state vectors (x) under bar (s). The results describe the algebraic and geometric dimensions of the invariant partitioning of the set of reachability indices of singular systems. The classification of all proper and nonproper polynomial vectors (x) under bar (s) induces a corresponding classification for the reachability spaces to proper-nonproper and results related to the possible dimensions feedback-spectra assignment properties of them are also given. The classification of minimal bases introduces new feedback invariants for singular systems, based on the real geometry of polynomial minimal bases, and provides an extension of the standard theory for proper systems (Warren, M.E., & Eckenberg, A.E. (1975).