The purpose of the present paper is to investigate the structure of distance spheres and cut locus $C(K)$ to a compact set $K$ of a complete Alexandrov surface $X$ with curvature bounded below. The structure of distance spheres around $K$ is almost the same as that of the smooth case. However $C(K)$ carries different structure from the smooth case. As is seen in examples of Alexandrov surfaces, it is proved that the set of all end points $C_e(K)$ of $C(K)$ is not necessarily countable and may possibly be a fractal set and have an infinite length. It is proved that all the critical values of the distance function to $K$ is closed and of Lebesgue measure zero. This is obtained by proving a generalized Sard theorem for one-valuable continuous functions. Our method applies to the cut locus to a point at infinity of a noncompact $X$ and to Busemann functions on it. Here the structure of all co-points of asymptotic rays in the sense of Busemann is investigated. This has not been studied in the smooth case.