abstract: Operators which exhibit both singular integral and generalized Radon transform singularities include SIO and fractional integral operators along families of curves or surfaces. Degenerate geometry of the curves or surfaces is reflected in degenerate microlocal geometry of the operators. Similar oscillatory operators come up in several inverse problems, and in those settings one is interested in both finding estimates and constructing parametrices. I will discuss a class of operators arising in seismology in the presence of conjugate points (caustics), but closely related to Hilbert transforms along curves. This is joint work with Raluca Felea and Malabika Pramanik.