  Motivated by the experimental identification of a higher-order van Hove
singularity (VHS) in AV$_3$Sb$_5$ kagome metals, we study electronic
instabilities of 2D lattice models with higher-order VHS and flavor degeneracy.
In contrast to conventional VHSs, the larger power-law density of states and
the weaker nesting propensity of higher-order VHSs conspire together to
generate distinct competing instabilities. After discussing the occurrence of
higher-order VHSs on square and kagome lattice models, we perform unbiased
renormalization group calculations to study competing instabilities and find a
rich phase diagram containing ferromagnetism, anti-ferromagnetism,
superconductivity and Pomeranchuk orders. Remarkably, there is a generic
transition from superconductivity to a $d$-wave Pomeranchuk order with
increasing flavor number. Implications for the intriguing quantum states of
AV$_3$Sb$_5$ kagome metals are also discussed.
