  In two-dimensional electronic lattices, changes in the topology of the Fermi
surface (Lifshitz transitions) lead to Van Hove singularities characterized by
a divergence in the electronic density of states. Van Hove singularities can
enhance the effect of electronic interactions, providing a platform to explore
novel correlated electronic states. In this work, we investigate the emergence
of topological Chern bands on the surface of three-dimensional topological
insulators, which host higher-order Van Hove singularities that are
characterized by the power-law diverging density of states. These singularities
can arise from the interplay between a time-reversal breaking Zeeman field
induced by proximity to a ferromagnetic insulator and a time-reversal invariant
moir\'e potential on the surface electrons, created by quintuple layer
misalignment in a family of topological insulators such as Bi$_2$Se$_3$ and
Bi$_2$Te$_3$, which host a single surface Dirac fermion. We establish the onset
of Chern bands near charge neutrality with Chern numbers $C = \pm 1$ that also
possess a manifold of higher-order Van Hove singularities on the moir\'e
Brillouin zone valleys controlled by the Zeeman and moir\'e potential energy
scales, unveiling a new platform to realize exotic Lifshitz transitions in
topological bands. Furthermore, we show that the strong peaks in the density of
states in the vicinity of Lifshitz transitions give rise to characteristic
features in the low-temperature intrinsic anomalous Hall conductivity, yielding
a path to probe Van Hove singularities in Chern bands through anomalous
transport measurements.
