  We study the presence of higher-order van Hove singularities in
mirror-symmetric twisted trilayer graphene. This geometry has recently emerged
experimentally as a fascinating playground for studying correlated and exotic
superconducting phases. We find that the trilayer hosts a zero-energy
higher-order van Hove singularity with an exponent -1/3. The singularity is
protected by the threefold rotation symmetry and a combined
mirror-particle-hole symmetry and it can be tuned with only the twist angle and
a perpendicular electric field. It arises from the combined merging of van Hove
singularities and Dirac cones at zero energy, beyond the recent classifications
of van Hove singularities. Moreover, we find that varying a third parameter
such as corrugation brings the system to a topological Lifshitz transition,
with anomalous exponent -2/5, separating regions of locally open and closed
semiclassical orbits.
