In this talk, I will show how one can extend the classical result of Alinhac and Baouendi to critically singular wave operators. In particular, we will consider wave operators blowing up critically on a spacelike or null hypersurface and will show that the unique continuation property from such a hypersurface does not hold by constructing counterexamples, provided there exists a family of trapped null geodesics. As an application to relativity and holography, I will also show how one can apply this non-uniqueness result to obtain counterexamples to unique continuation for some Klein-Gordon equations from the conformal boundaries of asymptotically Anti-de Sitter spacetimes. This work is in collaboration with Arick Shao.