Summary: Dielectric materials like electro-active polymers (EAPs) exhibit coupled electro-mechanical behavior at large strains. They respond by a deformation to an applied electrical field and are used in advanced industrial environments as sensors and actuators, for example, in robotics, biomimetics and smart structures. In field-activated or electronic EAPs, the electric activation is driven by Coulomb-type electrostatic forces, resulting in Maxwell stresses. These materials are able to provide finite actuation strains, which can even be improved by optimizing their composite microstructure. However, EAPs suffer from different types of instabilities. This concerns global structural instabilities, such as buckling and wrinkling of EAP devices, as well as local material instabilities, such as limit-points and bifurcation-points in the constitutive response, which induce snap-through and fine scale localization of local states. In this work, we outline variational-based definitions for structural and material stability, and design algorithms for accompanying stability checks in typical finite element computations. The formulation starts from stability criteria for a canonical energy minimization principle of electro-elasto-statics, and then shifts them over to representations related to an enthalpy-based saddle point principle that is considered as the most convenient setting for numerical implementation. Here, global structural stability is analyzed based on a perturbation of the total electro-mechanical energy, and related to statements of positive definiteness of incremental finite element tangent arrays. We base the local material stability on an incremental quasi-convexity condition of the electro-mechanical energy, inducing the positive definiteness of both the incremental electro-mechanical moduli as well as a generalized acoustic tensor. It is shown that the incremental arrays to be analyzed in the stability criteria appear within the enthalpy-based setting in a distinct diagonal form, with pure mechanical and pure electrical partitions. Applications of accompanying stability analyses in finite element computations are demonstrated by means of representative model problems.