We introduce and give the motivation for the concept of shell singularities. Shell singularities are a generalization of shell crossing singularities to spacetimes which are not necessarily spherically symmetric. We give a definition of strong cosmic censorship for shell singularities and show under which conditions shell singularities are censored. We study pregeodesics and geodesics which cross a shell singularity and introduce coordinates which are adapted to the singularity. We then specialize to spherical symmetry and discuss the family of perfect fluid spacetimes with shell crossing singularities given by Müller zum Hagen et al. We prove that spherically symmetric perfect fluid spacetimes with shell singularity have vanishing velocity of sound at the singularity. This lends further support to cosmic censorship.