In a recent work [Bret, EPL 135, 35001 (2021)10.1209/0295-5075/ac1e44], quantum electrodynamic (QED) effects were evaluated for the two-stream instability. It pertains to the growth of perturbations with a wave vector oriented along the flow in a collisionless counter-streaming system. Here, the analysis is extended to every possible orientation of the wave vector. The previous result for the two-stream instability is recovered, and it is proved that, even within the framework of a three-dimensional (3D) analysis, this instability remains fundamentally 1D even when accounting for QED effects. The filamentation instability, found for wave vectors normal to the flow, is weakly affected by QED corrections. As in the classical case, its growth rate saturates at large k_{⊥}. The saturation value is found independent of QED corrections. Also, the smallest unstable k_{⊥} is independent of QED corrections. Surprisingly, unstable modes found for oblique wave vectors do not follow the same pattern. For some, QED corrections do reduce the growth rate. But, for others, the same corrections increase the growth rate instead. The possibility for QED effects to play a role in unmagnetized systems is evaluated. Pair production resulting from γ emission by particles oscillating in the exponentially growing fields is not accounted for.