Underground energy storage is essential for the country’s development, and underground salt cavern groups are a productive way to store energy. Safety pillar design is the key to ensuring the safe operation of large salt cavern gas storage groups. Therefore, this paper is based on the salt pillar stability design problem and analyzes three aspects: (1) Three kinds of pillar stability design theories—reliability theory, strain energy theory, and catastrophe theory; (2) Two methods for designing stable pillars—determining pillars through formulas and numerical simulations; (3) The form and influencing factors of salt pillar instability—macro and microform and the main influential factors. From the current research, the major problem is that the design of safety pillars has not been systematic, due to the differences in salt rock in the region, and the engineering of salt pillar design still relies on experience. Finally, the impact of salt pillar width and gas injection and withdrawal on the pillars during the design of the salt cavern is analyzed, and the existing salt pillar theoretical characteristics and development trend are summarized.