Classical singularities in general relativity are usually defined as points of a "singular boundary" of space-time. However, this definition leads to serious difficulties if one applies it to stronger types of singularities, for instance to the initial and final singularities in the closed Friedman world model. It is shown that one can model space-time with any type of singularities as a noncommutative space. Moreover, any such singularity can be analysed in terms of an algebra of bounded operators on a Hilbert space. This typically quantum mechanical structure suggests that there could exist a deep connection between the theory of classical singularities and the looked-for quantum theory of gravity.