The notion of quantitative singular sets for spaces with lower Ricci curvature bounds was initiated by Cheeger and Naber. Volume estimates were proved for these singular sets in a non-collapsing setting. For Alexandrov spaces, we obtain stronger and volume-free estimates. We also show that the $(k,\epsilon)$-singular sets are $k$-rectifiable and such structure is sharp in some sense. This is a joint work with Aaron Naber.