This talk is devoted to the study of the fractal and topological properties of the sets of quasi-normal, partially, and essentially anormal numbers for the factorial numeral system. In particular, we prove that the set of essentially anormal numbers generated by the factorial expansion is residual. The superfractality of the set of partially anormal and essentially anormal numbers for the factorial expansion is also proved. On the other hand, in the talk, we present new properties about the calculation of the Hausdorff–Besicovitch dimension for subsets of the set of quasi-normal numbers for the factorial expansion. Possible generalizations of the obtained results and their transfer to other classes of the Cantor expansions are also discussed.