It is well known that on $\reals^n$, the Schrödinger propagator is unitary on $L2$ based spaces, but that locally in space and on average in time there is a $1/2$ derivative smoothing effect. We consider a family of manifolds with trapped geodesics which are degenerately hyperbolic and prove a sharp local smoothing estimate with loss depending on the type of trapping. Further, we construct a microlocal parametrix extended polynomially beyond Ehrenfest time, and as a consequence, we obtain Strichartz estimates with near-sharp loss depending only on the dimension of the trapping. This is partly joint work with J. Wunsch (Northwestern)