  The spin-Hall effect underpins some of the most active topics in modern
physics, including spin torques and the inverse spin-Hall effect, yet it lacks
a proper theoretical description. This makes it difficult to differentiate the
SHE from other mechanisms, as well as differentiate band structure and disorder
contributions. Here, by exploiting recent analytical breakthroughs in the
understanding of the intrinsic spin-Hall effect, we devise a density functional
theory method for evaluating the conserved (proper) spin current in a generic
system. Spin non-conservation makes the conventional spin current physically
meaningless, while the conserved spin current has been challenging to evaluate
since it involves the position operator between Bloch bands. The novel method
we introduce here can handle band structures with arbitrary degeneracies and
incorporates all matrix elements of the position operator, including the
notoriously challenging diagonal elements, which are associated with Fermi
surface, group velocity, and dipolar effects but often diverge if not treated
correctly. We apply this method to the most important classes of spin-Hall
materials: topological insulators, 2D quantum spin-Hall insulators,
non-collinear antiferromagnets, and strongly spin-orbit coupled metals. We
demonstrate that the torque dipole systematically suppresses contributions to
the conventional spin current such that, the proper spin current is generally
smaller in magnitude and often has a different sign. Remarkably, its
energy-dependence is relatively flat and featureless, and its magnitude is
comparable in all classes of materials studied. These findings will guide the
experiment in characterizing charge-to-spin interconversion in spintronic and
orbitronic devices. We also discuss briefly a potential generalisation of the
method to calculate extrinsic spin currents generated by disorder scattering.
