Thin membranes are notoriously sensitive to instabilities under mechanical loading, and need sophisticated analysis methods. Although analytical results are available for several special cases and assumptions, numerical approaches are normally needed for general descriptions of non-linear response and stability. The paper uses the case of a thin spherical hyper-elastic membrane subjected to internal gas over-pressure to investigate how stability conclusions are affected by chosen material models and kinematic discretizations. For spherical symmetry, group representation theory leads to linearized modes on the uniformly stretched sphere, with eigenvalues obtained from the mechanics of a thin membrane. A complete three-dimensional geometric description allows non-axisymmetric shear modes of the sphere, and such instabilities are shown to exist. When the symmetry of the continuous sphere is broken by discretized models, group representation theory gives predictions on the effects on the critical states. Numerical simulations of the pressurized sphere show and verify stability conclusions for sets of meshing strategies and hyper-elastic models.