The structure of the neutron-rich carbon isotopes ${}^{16,17,19}\mathrm{C}$ has been investigated using one-neutron knockout reactions on a ${}^{9}\mathrm{Be}$ target at approximately 60 MeV/nucleon. Partial cross sections and associated momentum distributions corresponding to final states of the ${}^{15,16,18}\mathrm{C}$ residues were measured and compared with predictions based on a shell-model theory and an eikonal model of the reaction mechanism. Spectroscopic factors and l-value assignments are given. The ground-state spins of ${}^{17,19}\mathrm{C}$ are ${\frac{3}{2}}^{+}$ and ${\frac{1}{2}}^{+},$ respectively. It is suggested that the accepted one-neutron separation energy for the ground state of ${}^{19}\mathrm{C}$ needs to be revised upwards.