Results are reported for single crystals of the PbFCl-type layered compound ${\mathrm{ZrP}}_{1.27}{\mathrm{Se}}_{0.73}$ that were produced using the iodine vapor transport method. Electrical transport, magnetization, and heat capacity measurements reveal disordered metallic behavior and the occurrence of bulk superconductivity, with a transition temperature $({T}_{c})$ of 7.1 K and an anisotropic orbitally limited upper critical field. $^{31}\mathrm{P}$ nuclear magnetic resonance measurements provide additional microscopic information, where the line shape, Knight shift, and spin-lattice relaxation data are consistent with the superconductivity originating from the corrugated Zr-P(2c)/Se plane. This suggests that the superconductivity first forms in the corrugated plane and the bulk superconductivity eventually occurs via coupling between the square planar layers. These data also show that either there are multiple superconducting gaps or there is a single gap that does not fully open across the entire Fermi surface. These results clarify the superconducting state in this material and will enable further measurements that require single-crystal specimens.