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Hybrid organic-inorganic perovskites (HOIPs) have seen promising developments in recent years in photovoltaics, light emitting diodes, and lasing applications. Understanding the photoluminescence (PL) and exciton properties of HOIPs plays a key role in optimizing their performance. PL is an important technique for the investigation of carrier-phonon interactions and quantification of defects in a system. However, these properties are inaccessible when important signatures are masked by the wide inhomogeneous broadening seen in polycrystalline films. Here, we study PL on MAPbI₃ single crystals in the low temperature orthorhombic phase. The high crystalline quality and well controlled defect density allow us to obtain PL spectra with narrow features and reveal previously unresolveable emissions. By comparing single crystals with low and moderate defect densities, the temperature and excitation intensity dependence data show evidence of the free exciton, bound excitons, and other defect dependent emissions in this material. The potential origin of these peaks will be discussed, as well as the measured exciton binding and exciton-phonon energies. These fine PL features will allow for further in-depth studies of the intrinsic photophysics and defect properties in HOIPs.