The angular distributions of Drell-Yan charged lepton pairs in the vicinity of the Z-boson mass peak probe the underlying QCD dynamics of Z-boson production. This paper presents a measurement of the complete set of angular coefficients A0−7 describing these distributions in the Z-boson Collins-Soper frame. The data analysed correspond to 20.3 fb−1 of pp collisions at s√=8s=8 TeV, collected by the ATLAS detector at the CERN LHC. The measurements are compared to the most precise fixed-order calculations currently available (O(α2s))(O(αs2)) and with theoretical predictions embedded in Monte Carlo generators. The measurements are precise enough to probe QCD corrections beyond the formal accuracy of these calculations and to provide discrimination between different parton-shower models. A significant deviation from the (O(α2s))(O(αs2)) predictions is observed for A0 − A2. Evidence is found for non-zero A5,6,7, consistent with expectations.