We present the first lattice QCD study of coupled-channel $D\pi$, $D\eta$ and $DsK¯$ scattering in isospin-1/2 in three partial waves. Using distillation, we compute matrices of correlation functions with bases of operators capable of resolving both meson and meson-meson contributions to the spectrum. These correlation matrices are analysed using a variational approach to extract the finite-volume energy eigenstates. Utilising L"uscher's method and its extensions, we constrain scattering amplitudes in $S$, $P$ and $D$-wave as a function of energy. By analytically continuing the scattering amplitudes to complex energies, we investigate the $S$-matrix singularities. Working at $m\pi \approx 391$ MeV, we find a pole corresponding to a $JP=0+$ near-threshold bound state with a large coupling to $D\pi$. We also find a deeply bound $JP=1-$ state, and evidence for a $JP=2+$ narrow resonance coupled predominantly to $D\pi$. Elastic $D\pi$ scattering in the isospin-$3/2$ channel is studied and we find a weakly repulsive interaction in $S$-wave.