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An improved measurement of the decay $B^0_s\to\mu^+\mu^-$ and searches forthe decays $B^0\to\mu^+\mu^-$ and $B^0_s\to\mu^+\mu^-\gamma$ are performed atthe LHCb experiment using data collected in proton-proton collisions at$\sqrt{s} = 7,~8$ and $13$ TeV, corresponding to integrated luminosities of 1,2 and 6 fb$^{-1}$, respectively. The $B^0_s\to\mu^+\mu^-$ branching fractionand effective lifetime are measured to be${\mathcal{B}}(B^0_s\to\mu^+\mu^-)=\left(3.09^{+0.46+0.15}_{-0.43-0.11}\right)\times10^{-9}$ and $\tau(B^0_s\to\mu^+\mu^-)=(2.07\pm 0.29\pm 0.03)$ ps,respectively, where the uncertainties include both statistical and systematiccontributions. No significant signal for $B^0\to\mu^+\mu^-$ and$B^0_s\to\mu^+\mu^-\gamma$ decays is found and the upper limits$\mathcal{B}(B^0\to\mu^+\mu^-)<2.6\times 10^{-10}$ and$\mathcal{B}(B^0_s\to\mu^+\mu^-\gamma)<2.0\times 10^{-9}$ at 95% confidencelevel are determined, where the latter is limited to the range $m_{\mu\mu} >4.9$ GeV$/c^2$. Additionally, the ratio between the $B^0\to\mu^+\mu^-$ and$B^0_s\to\mu^+\mu^-$ branching fractions is measured to be$\mathcal{R}_{\mu^+\mu^-}<0.095$ at 95% confidence level. The results are inagreement with the Standard Model predictions.

Measurement of the $B^0_s\to\mu^+\mu^-$ decay properties and search for the $B^0\to\mu^+\mu^-$ and $B^0_s\to\mu^+\mu^-\gamma$ decays