Devices that show a magnetic anisotropy normal to the film surface hold great promise towards faster and smaller magnetic bits in data-storage applications. We describe an experimental demonstration of current-induced magnetic reversal of nanopillars with perpendicular anisotropy and high coercive fields. The best results are observed for Co/Ni multilayers, which have higher giant magnetoresistance values and spin-torque efficiencies than Co/Pt multilayers. The reference layers were designed to have significantly higher anisotropy allowing a complete current-field phase diagram of the free-layer reversal to be explored. The results are compared to micromagnetic modelling of the free layer that, depending on the bias current and applied field, details regions of irreversible magnetic switching, coherent and incoherent spin waves, or static non-uniform magnetization states. This ability to manipulate high-anisotropy magnetic elements could prove useful for a range of spintronic applications.