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The main scheme in this report is concerned with the Morse-Sard theorem and Uniformly perfect sets.We mainly study the structure of singularity sets in the Morse-Sard theorem and the structure of uniformly perfect sets.Some results about these sets are given in this report.In chapter 1,we list some preliminaries and notations we need in the next chapters.In chapter 2,we give an account with discussed bibliography of what is known about the Morse-Sard theorem whether it is finite dimensional or not.We obtain a general version of Morse-Sard theorem and a counterexample in the infinite-dimensional case,a similar result is given in [BM]independently.In chapter 3,we discuss a set comes from complex dynamics:uniformly per-fect set.we consider Hausdorff dimension and Hausdorff measure about uniformly perfect sets,and have obtained some important results.