The solution to the Yamabe problem of finding a constant scalar curvature metric in a prescribed conformal class on a closed manifold was a major achievement in Geometric Analysis. Among several interesting generalizations to open manifolds, great attention has been devoted to the so-called "singular Yamabe problem". Given a closed Riemannian manifold M and a submanifold S, this problem consists of finding a complete metric on the complement of S in M that has constant scalar curvature and is conformal to the original metric. In other words, these are solutions to the Yamabe problem on M that blow up along S. A particularly interesting case is the one in which M is a round sphere and S is a great circle. In this talk, I will describe how bifurcation techniques and spectral theory of hyperbolic surfaces can be used to prove the existence of uncountably many nontrivial solutions to this problem. This is based on joint work with B. Santoro and P. Piccione.