We are concerned with the existence and qualitative properties of
travelling wave solutions for a quasilinear reaction-diffusion equation
on the real line. We consider a non-Lipschitz reaction term of
Fisher--KPP type and a discontinuous diffusion coefficient that allows
for degenerations and singularities at equilibrium points. We
investigate the joint influence of the reaction and diffusion terms on
the existence and nonexistence of travelling waves and, assuming these
terms are of power-type near equilibria, we provide classification of
solutions based on their asymptotic properties. Our approach provides a
broad theoretical background for the mathematical treatment of rather
general models not only in population dynamics but also in other applied
sciences and engineering.