  We investigate topological superconductivity in the Rashba-Hubbard model,
describing heavy-atom superlattice and van der Waals materials with broken
inversion. We focus in particular on fillings close to the van Hove
singularities, where a large density of states enhances the superconducting
transition temperature. To determine the topology of the superconducting gaps
and to analyze the stability of their surface states in the presence of
disorder and residual interactions, we employ an fRG+MFT approach, which
combines the unbiased functional renormalization group (fRG) with a real-space
mean-field theory (MFT). Our approach uncovers a cascade of topological
superconducting states, including $A_1$ and $B_1$ pairings, whose wave
functions are of dominant $p$- and $d$-wave character, respectively, as well as
a time-reversal breaking $A_1 + i B_1$ pairing. While the $A_1$ and $B_1$
states have first order topology with helical and flat-band Majorana edge
states, respectively, the $A_1 + i B_1$ pairing exhibits second-order topology
with Majorana corner modes. We investigate the disorder stability of the bulk
superconducting states, analyze interaction-induced instabilites of the edge
states, and discuss implications for experimental systems.
