Artificial intelligence (AI) as a multi-purpose technology is gaining increased attention and is now widely used across all sectors of the economy. The growing complexity of planning and operating power systems makes AI extremely valuable for the power industry. Until now, there has been a lack of clarity regarding the specific points along the power system supply chain where AI applications demonstrate significant value, as well as which AI domains are best suited for such applications. This study employs an AI taxonomy and automated web search to qualitatively and quantitatively unveil the biggest potentials of AI in the power industry. Our analysis, based on a review of 258'919 publications between 1982 and 2022, reveals where AI applications are particularly promising. We consider six AI domains (reasoning, planning, learning, communication, perception, integration & interaction) and 19 use cases from the power supply chain (i.e., generation, transmission networks, distribution networks, isolated grids/ microgrids, market operations and retail). Our findings indicate that, as of now, the focus is predominantly on AI applications in power retail (55 %), transmission (14 %) and generation (13 %). Most analyzed works describe applications built on algorithms of the AI domains "learning" (45 %) and "planning" (14 %). Results also suggest that the current definition of AI domains is ambiguous, and they highlight missing information on the actual use and successful implementation of AI in power system use cases.