In this paper, we study the superconvergence behavior of
discontinuous Galerkin methods using upwind numerical fluxes
for one-dimensional linear hyperbolic equations with degenerate
variable coefficients. The study establishes superconvergence results for the flux
function approximation as well as for the DG solution
itself. To be more precise, we first prove
that the DG flux function is superconvergent towards a
particular flux function of the exact solution, with an
order of $O(h^{k+2})$, when piecewise polynomials of degree $k$
are used. We then prove that the highest superconvergence
rate of the DG solution itself is $O(h^{k+\frac 32})$ as
the variable coefficient degenerates or achieves the value
zero in the domain. As byproducts, we obtain superconvergence properties for the
DG solution and the DG flux function at special
points and for cell averages.
All theoretical findings are confirmed by numerical experiments.