  Square-octagon lattice underlies the description of a family of
two-dimensional materials such as tetragraphene. In the present paper we show
that the tight-binding model of square-octagon lattice contains both
conventional and high-order van Hove points. In particular, the spectrum of the
model contains flat lines along some directions composed of high-order saddle
points. Their role is analyzed by calculating orbital susceptibility of
electrons. We find that the presence of van Hove singularities of different
kinds in the density of states leads to strong responses: paramagnetic for
ordinary singularities and more complicated for high-order singularities. It is
shown that the orbital susceptibility as a function of hoppings ratio $\alpha$
reveals the dia- to paramagnetic phase transition at $\alpha\approx 0.94$. This
is due to the competition of paramagnetic contribution of high-order VHS and
diamagnetic contribution of Dirac cones. The results for the tight-binding
model are compared with low-energy effective pseudospin-1 model near the three
band touching point.
