For example, in iterated prisoner's dilemma "always cooperate with your opponent" is not stable, because your opponent's optimal strategy against that is to defect every turn. The simulation I linked in my original comment shows a ton of strategies that are not stable and shows quite directly how they would quickly get eliminated by evolution.

For a simple example in evolution, most mutations harm the organism and are unstable. If most organism in a population had a very harmful mutation and a small population didn't, that small population would quickly take over the larger population. Hence, that mutation is unstable.

A slightly nontrivial example would be blind altruism in a situation where your species is severely starved of resources. If most animals were blindly altruistic and a small number of animals were not and would take advantage of the altruistic animals, then again, that small number would outcompete the larger population. So blind altruism isn't stable.

Of course we can't find many real-life examples; that is because they tend to be quickly eliminated by evolution. If they exist, it's usually only temporary.

Ah, wait, just in case... when I say "stability" it has nothing to do with stability of government and things like that. I meant it in more of the physics sense, that small perturbations wouldn't cause a large effect.