  We develop a detailed and self-consistent numerical model for
extremely-magnetised white dwarfs, which have been proposed as progenitors of
overluminous Type Ia supernovae. This model can describe fully-consistent
equilibria of magnetic stars in axial symmetry, with rotation,
general-relativistic effects and realistic equations of state (including
electron-ion interactions and taking into account Landau quantisation of
electrons due to the magnetic field). We study the influence of each of these
ingredients onto the white dwarf structure and, in particular, on their maximum
mass. We perform an extensive stability analysis of such objects, with their
highest surface magnetic fields reaching $\sim 10^{13}~G$ (at which point the
star adopts a torus-like shape). We confirm previous speculations that although
very massive strongly magnetised white dwarfs could potentially exist, the
onset of electron captures and pycnonuclear reactions may severely limit their
stability. Finally, the emission of gravitational waves by these objects is
addressed, showing no possibility of detection by the currently planned
space-based detector eLISA.

Solar and Stellar Astrophysics;High Energy Astrophysical Phenomena