In iterative schemes of identification and control one-of the particular and important choices to make is the choice for a model uncertainty structure, capturing the uncertainty concerning the estimated plant model. Structures that are used in the recent literature encompass e.g. gap metric uncertainty, coprime factor uncertainty, and the Vinnicombe gap metric uncertainty. In this paper, we study the effect of these choices by comparing the sets of controllers that guarantee robust stability for the different model uncertainty bounds. In general these controller sets intersect. However in particular cases the controller sets are embedded, leading to uncertainty structures that are favourable over others. In particular, when restricting the controller set to be constructed as metric-bounded perturbations around the present controller, the so-called double Youla parametrization provides a set of robustly stabilizing controllers that is larger than corresponding sets that are achieved by using any of the other uncertainty structures. This is particularly of interest in controller tuning problems.