Meeting the nanometre resolution promised by super-resolution microscopy techniques (pointillist: PALM, STORM, scanning: STED) requires stabilizing the sample drifts in real time during the whole acquisition process. Metal nanoparticles are excellent probes to track the lateral drifts as they provide crisp and photostable information. However, achieving nanometre axial super-localization is still a major challenge, as diffraction imposes large depths-of-fields. Here we demonstrate fast full three-dimensional nanometre super-localization of gold nanoparticles through simultaneous intensity and phase imaging with a wavefront-sensing camera based on quadriwave lateral shearing interferometry. We show how to combine the intensity and phase information to provide the key to the third axial dimension. Presently, we demonstrate even in the occurrence of large three-dimensional fluctuations of several microns, unprecedented sub-nanometre localization accuracies down to 0.7 nm in lateral and 2.7 nm in axial directions at 50 frames per second. We demonstrate that nanoscale stabilization greatly enhances the image quality and resolution in direct stochastic optical reconstruction microscopy imaging.