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On the basis of quasipotential method in quantum electrodynamics we calculate corrections of order $\alpha^5$ and $\alpha^6$ to hyperfine structure of $S$-wave energy levels of muonic deuterium. Relativistic corrections, effects of vacuum polarization in first, second and third orders of perturbation theory, nuclear structure and recoil corrections are taken into account. The obtained numerical values of hyperfine splitting $\Delta E^{hfs}(1S)=50.2814$ meV ($1S$ state) and $\Delta E^{hfs}(2S)=6.2804$~meV ($2S$ state) represent reliable estimate for a comparison with forthcoming experimental data of CREMA collaboration. The hyperfine structure interval $\Delta_{12}=8\Delta E^{hfs}(2S)-\Delta E^{hfs}(1S)={-0.0379}$~meV can be used for precision check of quantum electrodynamics prediction for muonic deuterium.