This paper concerns with the emerging power system stability issues, classification, and research prospects under a high share of renewables and power electronics. The decades-old traditional power system is undergoing a fast transition with two most prominent features: 1) high-penetration of renewable power generators, utilizing intermittent renewable sources such as wind and solar, and 2) high-penetration of power electronic devices in the generation e.g., wind turbine converters and solar power inverters, transmission e.g., flexible ac or dc transmission system converters, and distribution/utilization systems e.g., electric vehicle and microgrid. The development of modern power systems with dual high-penetrations, i.e., high-penetrations of renewables and power electronic devices, influences the power system dynamics significantly and causes new stability issues. This paper first overviews equipment-level features and system-level stability challenges introduced under the dual high-penetration scenario of the modern power system. Next, the impacts of emerging stability challenges on various aspects of the classical stability issues and classifications are highlighted. Under this context, this paper discusses the validity and limitations of the existing classical and extended power system stability classifications proposed by different IEEE/Cigre Working Groups. Furthermore, a new power system stability classification framework is proposed, which not only maintains the inherent logic of the classical classification but also provides wide coverage and future adaptability of the emerging stability issues. Finally, various classification-oriented research prospects in the power system stability domain are highlighted. •Discussion on the device- and system-level features of power systems with dual high-penetrations.•Emerging stability issues due to dual high-penetrations and their impact on stability classifications.•A critical review of existing classical and extended power system stability classifications of 2004 and 2020.•Highlighting the classification-oriented future research directions.