  Van Hove singularities (VHss) in the vicinity of the Fermi energy often play
a dramatic role in the physics of strongly correlated electron materials. The
divergence of the density of states generated by VHss can trigger the emergence
of new phases such as superconductivity, ferromagnetism, metamagnetism, and
density wave orders. A detailed understanding of the electronic structure of
these VHss is therefore essential for an accurate description of such
instabilities. Here, we study the low-energy electronic structure of the
trilayer strontium ruthenate Sr$_4$Ru$_3$O$_{10}$, identifying a rich hierarchy
of VHss using angle-resolved photoemission spectroscopy and millikelvin
scanning tunneling microscopy. Comparison of $k$-resolved electron spectroscopy
and quasiparticle interference allows us to determine the structure of the VHss
and demonstrate the crucial role of spin-orbit coupling in shaping them. We use
this to develop a minimal model from which we identify a new mechanism for
driving a field-induced Lifshitz transition in ferromagnetic metals.
