  Due to the diverging density of states (DOS), Van Hove singularities (VHS)
near the Fermi level are known to boost the susceptibility to a wide variety of
electronic instabilities, including superconductivity. We theoretically show
that the number of VHS in the normal state can be qualitatively inferred from
the optical absorption spectra Re $\sigma_{ii}(\omega)$ in the superconducting
state. The key feature is the absorption peak at frequency $\omega=2\Delta$
from the optical transition across the superconducting gap $\Delta$, which is
forbidden in a single-band clean superconductor when the inversion symmetry is
preserved and the band is quadratic. Although VHS dispersions are mostly
quadratic, we find that a divergent peak occurs when there are multiple VHS on
the Fermi surface under an applied current. In contrast, we find non-diverging
weak peaks when there is only a single VHS. Depending on whether this single
VHS has logarithmically or power-law divergent DOS, the peaks in Re
$\sigma_{xx}(\omega)$ and Re $\sigma_{yy}(\omega)$ are nearly isotropic and
anisotropic, respectively. Therefore, we propose that experimentally measured
peak magnitude and anisotropy in the optical absorption spectra of VH-driven
superconductors can be used to determine the number and type of VHS on the
Fermi surface. Disorders could even facilitate in distinguishing the multiple
and single VHS scenarios since only the diverging peak in the latter case is
expected to survive.
