  The rare decay $B^0_s \to \mu^+\mu^-$ plays a key role for the testing of the
Standard Model. It is pointed out that the sizable decay width difference
$\Delta\Gamma_s$ of the $B_s$-meson system affects this channel in a subtle
way. As a consequence, its calculated Standard Model branching ratio has to be
upscaled by about 10%. Moreover, the sizable $\Delta\Gamma_s$ makes a new
observable through the effective $B^0_s\to \mu^+\mu^-$ lifetime accessible,
which probes New Physics in a way complementary to the branching ratio and adds
an exciting new topic to the agenda for the high-luminosity upgrade of the LHC.
Further probes of New Physics are offered by a CP-violating rate asymmetry.
Correlations between these observables and the $B^0_s \to \mu^+\mu^-$ branching
ratio are illustrated for specific models of New Physics.
