  The rare flavour-changing neutral current decay $B_s\to\mu^+\mu^-$ is among
the most important indirect probes of new physics at the LHC, as it is strongly
suppressed in the Standard Model, very sensitive to new physics effects, and
theoretically exceptionally clean. We present a thorough state-of-the-art
analysis of the constraints on new physics from present and future measurements
of this decay, focusing on scalar operators. We show model-independently and in
concrete new physics models, namely the MSSM and two leptoquark scenarios, that
a future precise measurement of the mass-eigenstate rate asymmetry in
$B_s\to\mu^+\mu^-$ would allow to disentangle new physics scenarios that would
be indistinguishable based on measurements of the branching ratio alone. We
also highlight the complementarity between $B_s\to\mu^+\mu^-$ and direct
searches in both model classes. Our numerics is based on the open source code
flavio and is made publicly available.
