The wrinkling instabilities of a pre-tensioned annular thin film undergoing azimuthal shearing are investigated within the framework of the linearized Donnell-von Karman bifurcation equation for thin plates. The main objective here is to provide a rational understanding of the role played by the presence of finite bending stiffness and to explain the nature of the localized deformation patterns observed in experiments. In order to achieve this, the eigenvalue problem is formulated as a differential equation with variable coefficients depending on a large parameter. The singular perturbation nature of this equation arises from a combination involving both the pre-stress and the geometrical features of the annular domain. The localization mechanism of the corresponding eigenmodes is then unravelled with the help of a WKB analysis motivated by the qualitative behavior of the neutral stability curves. We show that the asymptotic findings are in very good agreement with the results of direct numerical simulations of the original bifurcation equation.