I will discuss the connection between renormalization and rigidity of circle maps and spectral theory of Schroedinger operators over them. In particular, we consider Schroedinger operators over circle maps (with an invariant measure $\mu$) with several critical or break points, i.e. piecewise smooth homeomorphisms of a circle with several points where the derivatives vanish or have jump discontinuities. We show that, for $\mu$-almost all x, in a two-parameter region --- determined by the geometry of the dynamical partitions and $\alpha$ --- the spectrum of Schroedinger operators H(T,V,x) over every sufficiently smooth such map T, is purely singular continuous, for every $\alpha$-H\"older-continuous potential V.