- In this paper we prove that the spectrum of the Neumann-Lap\-la\-ci\-an (the free membrane problem) in a large class of rough space domains is discrete. We also obtain lower estimates for the first non-trivial eigenvalue, using the $p$-quasiconformal mappings theory. In our opinion the mainly applicable class for the spectral problem is a class of $2$-quasiconformal mappings. For planar domains this class coincides with the class of quasiconformal mappings. Proofs are based on the composition operators theory for Sobolev spaces with first weak derivatives.