  In this article we study conjectures regarding normalized volume and
boundedness of singularities. We focus on singularities with a torus action of
complexity 1, threefold singularities, and hypersurface singularities. Given a
real value v>0, we prove that the class of K-semistable threefold singularities
with normalized volume at least v forms a bounded family. Analogous statements
are proved in the case of n-dimensional complexity-1 and n-dimensional
hypersurface singularities for arbitary n. In the general case of klt
singularities, i.e. without the assumption on K-semistability, we show that, up
to special degenerations, the normalized volume bounds singularities with a
complexity-1 torus action. We exhibit a 3-dimensional example which shows that
this last statement is optimal.
