The present work describes the results of the removal of cesium by sorbents of various classes from highly mineralized alkaline solutions simulating the clarified phase of storage tanks with high-level radioactive waste (HLW) of the Mayak Production Association. Within the scope of the performed works, inorganic sorbents of the Clevasol® and Fersal brands, as well as resorcinol-formaldehyde ion-exchange resins (RFRs: RFR-i, RFR-Ca, and Axionit RCs), were used. The sorbents’ characteristics under both static and dynamic conditions are presented. The Fersal sorbent has demonstrated the best sorption characteristics in the series of sorbents under study. The disadvantage of inorganic sorbents is the loss of mechanical strength upon cesium desorption, which complicates their repeated use. It has been demonstrated that RFRs, despite their lower selectivity towards cesium and adsorption capacity, can be used many times in repeated sorption-desorption cycles. The latter makes RFRs more technologically attractive in terms of the total volume of decontaminated HLW. However, RFRs tend to be oxidized during storage, which results in the formation of carboxyl groups and a decrease in sorption characteristics—this must be further taken into account in the real processes of liquid radioactive waste (LRW) management.