A search for the rare decays $B^0_stomu^+mu^-$ and $B^0tomu^+mu^-$ is performed at the LHCb experiment using data collected in $pp$ collisions corresponding to a total integrated luminosity of 4.4 fb$^{-1}$. An excess of $B^0_stomu^+mu^-$ decays is observed with a significance of 7.8 standard deviations, representing the first observation of this decay in a single experiment. The branching fraction is measured to be ${cal B}(B^0_stomu^+mu^-)=left(3.0pm 0.6^{+0.3}_{-0.2}right)times 10^{-9}$, where the first uncertainty is statistical and the second systematic. The first measurement of the $B^0_stomu^+mu^-$ effective lifetime, $tau(B^0_stomu^+mu^-)=2.04pm 0.44pm 0.05$ ps, is reported. No significant excess of $B^0tomu^+mu^-$ decays is found and a 95 % confidence level upper limit, ${cal B}(B^0tomu^+mu^-)<3.4times 10^{-10}$, is determined. All results are in agreement with the Standard Model expectations.
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