The inorganic scintillator is an important element of a new type of cryogenic phonon scintillation detectors (CPSD) developed for single particle detection. These detectors exhibiting superior energy resolution and ability to identify the type of interaction event are considered as a next generation instrumentation in the search for extremely rare events. The paper presents the latest results of our study on cryogenic scintillators thought for CPSD application in the search for dark matter. It gives a description of the concept of direct dark matter detection and the operation principles of GPSD. The paper envisages major materials requirements and summarises the results of the studies of luminescence and scintillation properties of tungstates (CaWO 4 and ZnWO4), molybdates (CaMoO4, MgMoO 4, and CdMoO4) and Ti-doped Al2O3 over a wide temperature range (9-300 K).