We prove global and local weighted stability theorems for representations of abelian semigroups on Banach spaces under countable spectral conditions and certain growth assumptions on the weights. In the global case, we use a limit semigroup construction together with an extension theorem for weighted semigroup representations. In the local case, we adapt a definition of Albrecht to introduce a local spectrum of a representation which is no larger than the usual notion of local spectrum for representations of Z+ and R+, and we establish the corresponding local stability theorem. We give an application to the asymptotic theory of functions on abelian semigroups.