The rising demand for green energy to reduce carbon emissions is accelerating the integration of renewable energy sources (RESs) like wind and solar power. However, this shift presents significant challenges due to the inherent variability and intermittency of RESs, which impact power system stability and reliability. As a result, there is a growing need for enhanced flexibility to maintain stable and reliable operations. This study reviews recent advancements in power system flexibility enhancement, particularly concerning the integration of RESs, with a focus on the critical role of energy storage systems (ESSs) in mitigating these challenges. ESSs play a vital role in addressing the variability of RESs, supporting grid stability, and enabling energy consumption time shifting. While existing studies discuss power system flexibility, each typically addresses specific aspects. This comprehensive review evaluates flexibility measures for renewable-based electricity in terms of reliability and stability, highlighting the importance of ESSs in transmission, distribution networks, and end-user applications like residential buildings and vehicle-to-grid technologies to enhance overall system flexibility. Additionally, we identify recent challenges, such as the increased risk of grid congestion, frequency deviations, and the need for real-time supply and demand balancing, which necessitate innovative ESS applications. We propose future directions, including a transition pathway to promote the large-scale deployment of diverse ESS technologies to support grid modernization, enhance resilience, and foster sustainable power supply development. Our findings emphasize the growing research into optimizing power system stability and reliability, offering valuable guidance for future research and practical implementation.