Achieving a better understanding of the physics of defects in halide perovskites is a key challenge for improving the efficiency of the devices. Here, we present a comprehensive study of the defect emission of CH3NH3PbI3 perovskite single crystals. The emission of the pristine surface of cleaved crystals is systematically investigated based on steady-state and time-resolved micro-photoluminescence spectroscopy. Donor-acceptor pair (DAP) recombination is observed due to the presence of native shallow defects. The DAP spectra present an important variability depending on the location on the surface of the crystals due to inhomogeneous defect distribution. A strong blue-shift of the emission is measured as a function of excitation power and is explained by fluctuating potential caused by compensated defects. With increasing photocarrier density, a transition from a structureless to a structured DAP emission with several longitudinal optical (LO) phonon replicas is observed. The DAP transition is characterized by the red-shift of the emission with time and a slow, non-exponential photoluminescence decay. Sharp discrete lines with sub-meV widths, the most recognizable spectral signature of DAP transition, are revealed. The results reconcile apparently contradictory previous observations on defect emission of CH3NH3PbI3 and provide new insights into the properties of defects in halide perovskites.