The global demand for energy and the need to mitigate climate change require a shift from traditional fossil fuels to sustainable and renewable energy alternatives. Hydrogen is recognized as a significant component for achieving a carbon-neutral economy. This comprehensive review examines the underground hydrogen storage and, particularly, laboratory-scale studies related to rock–hydrogen interaction, exploring current knowledge. Using bibliometric analysis of data from the Scopus and Web of Science databases, this study reveals an exponential increase in scientific publications post-2015, which accounts for approximately 85.26% of total research output in this field and the relevance of laboratory experiments to understand the physicochemical interactions of hydrogen with geological formations. Processes in underground hydrogen storage are controlled by a set of multi-scale parameters, including solid properties (permeability, porosity, composition, and geomechanical properties) and fluid properties (liquid and gas density, viscosity, etc.), together with fluid–fluid and solid–fluid interactions (controlled by solubility, wettability, chemical reactions, etc.). Laboratory experiments aim to characterize these parameters and their evolution, simulating real-world storage conditions to enhance the reliability and applicability of findings. The review emphasizes the need to expand research efforts globally to comprehensively address the currently existing issues and knowledge gaps.