We study pneumatically inflated membranes indented by rigid indenters of different sizes and shapes. When the volume of the inflated membrane is beyond a critical value, a symmetric deformation mode becomes unstable and the system follows a path of asymmetric deformation. This bifurcation is analyzed analytically for a 2D membrane with either a line or plane indenter for which the stable deformation path is determined by computing the total system potential energy of different configurations. An axisymmetric membrane with indenters of different shapes and sizes is further investigated numerically. In this case, a cylindrical indenter can always trigger bifurcation while a small spherical indenter tends to be encapsulated rather than induce an asymmetric deformation mode. This result suggests that the observed bifurcation behavior can be actively tuned and even triggered selectively by tuning indenter shape and size. We also demonstrate the effects of friction and biased bifurcation analytically through the example of a 2D membrane with a line indenter.