Three-body decays of top and bottom squarks

24 pages with 4 psfigures, latexInternational audienceWe investigate the decays of third generation scalar quarks in the Minimal Supersymmetric extension of the Standard Model, focusing on the three-body modes. We calculate the partial widths of the decays of heavier top and bottom squarks into the lighter ones and a fermion pair [through virtual vector boson, Higgs boson or gaugino exchanges] and the partial widths of the three-body decays of both top squarks into bottom quarks and a pair of fermion and scalar fermion [we consider the case of lighter $\tilde{\tau}$ or $\tilde{b}$ states] and into a bottom quark, the lightest neutralino and a $W$ or a charged Higgs boson $H^\pm$. Some of these decay modes are shown to have substantial branching ratios in some areas of the parameter space