We report on a search for second generation scalar leptoquarks with the DO detector at the Fermilab Tevatron $p\overline{p}$ collider at $\sqrt{s}=1.8$ TeV. This search is based on 12.7 ${\mathrm{pb}}^{\ensuremath{-}1}$ of data. Second generation leptoquarks are assumed to be produced in pairs and to decay into a muon and quark with branching ratio $\ensuremath{\beta}$ or into a neutrino and quark with branching ratio $1\ensuremath{-}\ensuremath{\beta}$. We obtain cross section times branching ratio limits as a function of leptoquark mass and set lower limits on the leptoquark mass at the 95% confidence level.