  Topologically nontrivial two-dimensional materials hold great promise for
next-generation optoelectronic applications. However, measuring the Hall or
spin-Hall response is often a challenge and practically limited to the ground
state. An experimental technique for tracing the topological character in a
differential fashion would provide useful insights. In this work, we show that
circular dichroism angle-resolved photoelectron spectroscopy (ARPES) provides a
powerful tool which can resolve the topological and quantum-geometrical
character in momentum space. In particular, we investigate how to map out the
signatures of the local Berry curvature by exploiting its intimate connection
to the orbital angular momentum. A spin-resolved detection of the
photoelectrons allows to extend the approach to spin-Chern insulators. Our
predictions are corroborated by state-of-the art \emph{ab initio} simulations
employing time-dependent density functional theory, complemented with model
calculations. The present proposal can be extended to address topological
properties in materials out of equilibrium in a time-resolved fashion.
