  We generalize the classical study of Alexander polynomials of smooth or PL
locally-flat knots to PL knots that are not necessarily locally-flat. We
introduce three families of generalized Alexander polynomials and study their
properties. For knots with point singularities, we obtain a classification of
these polynomials that is complete except for one special low-dimensional case.
This classification extends existing classifications for PL locally-flat knots.
For knots with higher-dimensional singularities, we further extend the
necessary conditions on the invariants. We also construct several varieties of
singular knots to demonstrate realizability of certain families of polynomials
as generalized Alexander polynomials. These constructions, of independent
interest, generalize known knot constructions such as frame spinning and twist
spinning.
