In this work we investigate the possible development of instabilities in a certain class of dielectric-elastomer composites (DECs) subjected to all-around dead electromechanical loading. The DECs consist of a dielectric elastomer matrix phase constrained to plane strain deformations by means of aligned, long, rigid dielectric fibers of elliptical cross-section that are also aligned but randomly distributed in the transverse plane. Two types of instabilities are considered: loss of positive definiteness (LPD), and loss of strong ellipticity (LE). LPD simply corresponds to the loss of local convexity of the homogenized electroelastic stored-energy function for the DECs and can be of two types depending on the resulting instability modes. When the modes are aligned with the ‘principal’ solution, the instability corresponds to a maximum in the nominal electric field, possibly followed by snapping behavior. Alternatively, when the modes are orthogonal to the principal solution, the instability corresponds to a bifurcation from the principal solution. The LE, on the other hand, corresponds to LPD of the electromechanical acoustic tensor and manifests itself by the onset of highly localized shear band instabilities. Our results show that the stability, as well as the type of instability (when stability is lost), of the DECs depends sensitively on the loading conditions and is also affected by the microstructure of the DECs (i.e. the volume fraction of the fibers and their aspect ratio). In particular, it is found that the non-aligned LPD instabilities typically precede aligned LPD and LE instabilities, especially for in-plane isotropic microstructures, but aligned (limit load) LPD and LE instabilities are also possible for composites with anisotropic microstructures. In this context, it should be noted that the possible development of non-aligned LPD bifurcation instabilities appears to have been ignored by prior work. Fortunately, however, such instabilities can usually be avoided by orienting the long (in-plane) axis of the fibers parallel to the tensile stress direction and orthogonal to the applied electric field.