Abstract: In many important geometric PDEs, solutions are not necessarily regular. When the singularity occurs, we still intend to explore whether solutions possess regularity of a certain degree. The structure of the singular sets, where solutions become singular, play an important role. In this talk, we focus on two cases: the singular sets are isolated or the boundary of domains where solutions are defined. To study the regularity of solutions near the singular sets, we investigate asymptotic expansions of solutions. The talk is intended for a general audience.