The growing demand for high-performance, safe, and sustainable energy storage systems has spurred significant interest in advanced lithium-ion batteries (LIBs). While conventional liquid electrolytes have enabled commercial success, their flammability, leakage risk, and poor electrochemical stability continue to pose critical limitations. Solid-state electrolytes offer a compelling alternative, with solid organic electrolytes (SOEs) emerging as particularly promising due to their lightweight nature, mechanical flexibility, and chemical tunability. This review presents a comprehensive analysis of recent developments in SOEs for LIBs, with a focus on solid polymer electrolytes (SPEs) and single-ion conducting polymer electrolytes (SICPEs). We highlight key advances in polymer matrix design, the role of functional fillers, and the influence of anionic group chemistry—such as carboxylate, sulfonate, and sulfonylimide moieties—on ionic conductivity and transport mechanisms. In addition, we evaluate state-of-the-art fabrication techniques, including solution casting, in situ polymerization, phase inversion, and electrospinning, that are critical to achieving scalable and robust electrolyte membranes. By comparing organic and hybrid organic–inorganic strategies, this review outlines current challenges and future directions for SOE development aimed at realizing safe, efficient, and high-energy-density LIBs.