  Results are reported for the B$^0_\mathrm{s}\to\mu^+\mu^-$ branching fraction
and effective lifetime and from a search for the decay B$^0\to\mu^+\mu^-$. The
analysis uses a data sample of proton-proton collisions accumulated by the CMS
experiment in 2011, 2012, and 2016, with center-of-mass energies (integrated
luminosities) of 7 TeV (5 fb$^{-1}$), 8 TeV (20 fb$^{-1}$), and 13 TeV (36
fb$^{-1}$). The branching fractions are determined by measuring event yields
relative to B$^+\to$ J/$\psi$K$^+$ decays (with J/$\psi\to\mu^+\mu^-$), which
results in the reduction of many of the systematic uncertainties. The decay
B$^0_\mathrm{s}\to\mu^+\mu^-$ is observed with a significance of 5.6 standard
deviations. The branching fraction is measured to be
$\mathcal{B}$(B$^0_\mathrm{s}\to\mu^+\mu^-$) = [2.9 $\pm$ 0.7 (exp) $\pm$ 0.2
(frag)] $\times$ 10$^{-9}$, where the first uncertainty combines the
experimental statistical and systematic contributions, and the second is due to
the uncertainty in the ratio of the B$^0_\mathrm{s}$ and the B$^+$
fragmentation functions. No significant excess is observed for the decay
B$^0\to\mu^+\mu^-$, and an upper limit of $\mathcal{B}$(B$^0\to\mu^+\mu^-$) <
3.6 $\times$ 10$^{-10}$ is obtained at 95% confidence level. The
B$^0_\mathrm{s}\to\mu^+\mu^-$ effective lifetime is measured to be
$\tau_{\mu^+\mu^-}$ = 1.70 $^{+0.61}_{-0.44}$ ps. These results are consistent
with standard model predictions.
