Quite recently, examples of non-linear adaptive controllers have been found that adaptively stabilize any scalar first-order linear system. In this paper we describe two general classes of such ‘universal’ adaptive stabilizers that include the previously proposed controllers as special cases, Different types of perturbations of these controllers are studied and we report on extensive simulation experiments. Our results demonstrate considerable qualitative differences in the dynamical behaviour of various adaptive universal stabilizers. While the previously proposed controllers all exhibit the bursting phenomena, we give an explicit example of a stabilizing controller that does not show bursting behaviour.