  We report the evolution of the electronic structure at the surface of the
layered perovskite Sr$_2$RuO$_4$ under large in-plane uniaxial compression,
leading to anisotropic $B_{1g}$ strains of
${\varepsilon_{xx}-\varepsilon_{yy}=-0.9\pm0.1\%}$. From angle-resolved
photoemission, we show how this drives a sequence of Lifshitz transitions,
reshaping the low-energy electronic structure and the rich spectrum of van Hove
singularities that the surface layer of Sr$_2$RuO$_4$ hosts. From comparison to
tight-binding modelling, we find that the strain is accommodated predominantly
by bond-length changes rather than modifications of octahedral tilt and
rotation angles. Our study sheds new light on the nature of structural
distortions at oxide surfaces, and how targeted control of these can be used to
tune density of states singularities to the Fermi level, in turn paving the way
to the possible realisation of rich collective states at the Sr$_2$RuO$_4$
surface.
