  We show that in three-dimensional (3D) topological metals, a subset of the
van Hove singularities of the density of states sits exactly at the transitions
between topological and trivial gapless phases. We may refer to these as
topological van Hove singularities. By investigating two minimal models, we
show that they originate from energy saddle points located between Weyl points
with opposite chiralities, and we illustrate their topological nature through
their magnetotransport properties in the ballistic regime. We exemplify the
relation between van Hove singularities and topological phase transitions in
Weyl systems by analyzing the 3D Hofstadter model, which offers a simple and
interesting playground to consider different kinds of Weyl metals and to
understand the features of their density of states. In this model, as a
function of the magnetic flux, the occurrence of topological van Hove
singularities can be explicitly checked.
