  The properties of correlated electron materials are often intricately linked
to Van Hove singularities (VHs) in the vicinity of the Fermi energy. The class
of these VHs is of great importance, with higher order ones -- with power-law
divergence in the density of states -- leaving frequently distinct signatures
in physical properties. We use a new theoretical method to detect and analyse
higher order Van Hove singularities (HOVHs) in two-dimensional materials and
apply it to the electronic structure of the surface layer of Sr$_2$RuO$_4$. We
then constrain a low energy model of the VHs of the surface layer of
Sr$_2$RuO$_4$ against angle-resolved photoemission spectroscopy and
quasiparticle interference data to analyse the VHs near the Fermi level. We
show how these VHs can be engineered into HOVHs.
