  We present the first lattice QCD study of coupled-channel $D\pi$, $D\eta$ and
$D_{s}\bar{K}$ 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 $J^{P} = 0^{+}$
near-threshold bound state with a large coupling to $D\pi$. We also find a
deeply bound $J^{P} = 1^{-}$ state, and evidence for a $J^{P} = 2^{+}$ 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.
