We study the asymptotics of strongly continuous operator semigroups defined on locally convex spaces in order to develop a stability theory for solutions of evolution equations beyond Banach spaces. In the classical case, there is only little choice for a semigroup’s speed in approaching zero uniformly. Indeed, if a strongly continuous semigroup on a Banach space converges to zero uniformly at any speed then it converges already uniformly at exponential speed. Semigroups with this property are said to be exponentially stable. Leaving the Banach space setting, the situation changes entirely; for instance convergence to zero at a speed faster than any polynomial but not exponentially fast is possible. In this article we establish concepts of stability which refine the classical notions and allow to grasp the different kinds of asymptotic behavior. We give characterizations of the new properties, study their relations and consider generic examples like multiplication semigroups and shifts. In addition we apply our results to the transport and the heat equation on classical Fréchet function spaces.