A measurement of the differential branching fraction of the decay
${B^{0}\rightarrow K^{\ast}(892)^{0}\mu^{+}\mu^{-}}$ is presented together with
a determination of the S-wave fraction of the $K^+\pi^-$ system in the decay
$B^{0}\rightarrow K^{+}\pi^{-}\mu^{+}\mu^{-}$. The analysis is based on
$pp$-collision data corresponding to an integrated luminosity of 3\,fb$^{-1}$
collected with the LHCb experiment. The measurements are made in bins of the
invariant mass squared of the dimuon system, $q^2$. Precise theoretical
predictions for the differential branching fraction of $B^{0}\rightarrow
K^{\ast}(892)^{0}\mu^{+}\mu^{-}$ decays are available for the $q^2$ region
$1.1<q^2<6.0\,{\rm GeV}^2/c^4$. In this $q^2$ region, for the $K^+\pi^-$
invariant mass range $796 < m_{K\pi} < 996\,{\rm MeV}/c^2$, the S-wave fraction
of the $K^+\pi^-$ system in $B^{0}\rightarrow K^{+}\pi^{-}\mu^{+}\mu^{-}$
decays is found to be \begin{equation*}
F_{\rm S} = 0.101\pm0.017({\rm stat})\pm0.009 ({\rm syst}), \end{equation*}
and the differential branching fraction of $B^{0}\rightarrow
K^{\ast}(892)^{0}\mu^{+}\mu^{-}$ decays is determined to be \begin{equation*}
{\rm d}\mathcal{B}/{\rm d} q^2 = (0.342_{\,-0.017}^{\,+0.017}({\rm
stat})\pm{0.009}({\rm syst})\pm0.023({\rm norm}))\times 10^{-7}c^{4}/{\rm
GeV}^{2}. \end{equation*} The differential branching fraction measurements
presented are the most precise to date and are found to be in agreement with
Standard Model predictions.