Some properties that I notice about semistable equilibria:
It is non-differentiable, so any semsistable equilibrium that occurs in reality is only approximate.
If the zone of attraction and repulsion are the same state, random noise will inevitably cause the state to hop over to the repulsive side. So what a ‘perfect’ semistable equilibrium will look like is a system where the state tends towards some point, hangs around for a while, and then suddenly flies off to the next equilibrium. This makes me think of the Gömböc.
A more approximate semsistable equilibrium that has an actual stable point in reality will be one that has a stable equilibrium at one point, and an unstable equilibrium soon after. I think an example of this is a neutron star. A neutron star is stable because gravity pulls the matter inward while the nuclear forces push outward. With more compression however, gravity overcomes these forces and a black hole forms, after which the entire star will collapse.