  The saddle point (van Hove singularity) exhibits a divergent density of
states in 2D systems, leading to fascinating phenomena like strong correlations
and unconventional superconductivity, yet it is seldom observed in 3D systems.
In this work, we have found two types of 3D higher-order saddle points (HOSPs)
in emerging 3D kagome metals, YbCo$_6$Ge$_6$ and MgCo$_6$Ge$_6$. Both HOSPs
exhibit a singularity in their density of states, which is significantly
enhanced compared to the ordinary saddle point. The HOSP near the Fermi energy
generates a flat band extending a large area in the Brillouin zone, potentially
amplifying the correlation effect and fostering electronic instabilities. Two
types of HOSPs exhibit distinct robustness upon element substitution and
lattice distortions in these kagome compounds. Our work paves the way for
engineering exotic band structures, such as saddle points and flat bands, and
exploring interesting phenomena in Co-based kagome materials.
