A measurement of the differential branching fraction of the decay B 0 → K ∗(892)0 μ + μ − is presented together with a determination of the S-wave fraction of the K + π − system in the decay B 0 → K +π− μ + μ −. 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 → K ∗(892)0 μ + μ − decays are available for the q 2 region 1.1 < q 2 < 6.0 GeV2 /c 4. In this q 2 region, for the K +π− invariant mass range 796 < m Kπ < 996 MeV/c 2, the S-wave fraction of the K +π− system in B 0 → K +π− μ + μ − decays is found to be $$ {F}_{\mathrm{S}}=0.101\pm 0.017\left(\mathrm{stat}\right)\pm 0.009\left(\mathrm{syst}\right), $$ and the differential branching fraction of B 0 → K ∗(892)0 μ + μ − decays is determined to be $$ \mathrm{d}\mathrm{\mathcal{B}}/\mathrm{d}{q}^2=\left(0.{392}_{-0.019}^{+0.020}\left(\mathrm{stat}\right)\pm 0.010\left(\mathrm{syst}\right)\pm 0.027\left(\mathrm{norm}\right)\right)\times 1{0}^{-7}{c}^4/{\mathrm{GeV}}^2. $$ The differential branching fraction measurements presented are the most precise to date and are found to be in agreement with Standard Model predictions.