We have performed low-temperature photoluminescence and photoluminescence excitation (PLE) measurements on highly degenerate p-type GaAs and InyGa1-yAs quantum wells. In the PLE spectrum of the GaAs well, evidence of a second-order van Hove singularity in the joint density of states of the ground-state light-hole and electron bands is found. This singularity results from the equality of ground-state light-hole and electron effective masses near the Gamma point, being a much more restrictive demand than the usual condition for a van Hove singularity, which requires only the equality of first derivatives of the subband dispersions. The second-order van Hove singularity gives rise to a power-law divergence at the singular point, whereas the corresponding usual van Hove singularity results in a steplike discontinuity in the joint density of states. The observed singularity could be described extremely well by a simple analytical model. The increased energy gap between light- and heavy-hole ground states in the compressively strained InyGa1-yAs well enhances the valence-band parabolicity, resulting in the disappearance of the van Hove singularity. Furthermore, it is shown that the anisotropic character of the heavy-hole ground state in GaAs is strongly suppressed in the InyGa1-yAs system. All experiments are in good agreement with our numerical modeling, based on an exact solution of the 4 x 4 Luttinger Hamiltonian.