We propose an algorithm (SCS-FRI) to estimate multipath channels with Sparse Common Support (SCS) based on Finite Rate of Inno- vation (FRI) sampling. In this setup, theoretical lower-bounds are derived, and simulation in a Rayleigh fading environment shows that SCS-FRI gets very close to these bounds. We show how to apply SCS-FRI to OFDM and CDMA downlinks. Recovery of a sparse common support is, among other, especially relevant for channel es- timation in a multiple output system or beam-forming from multiple input. The present algorithm is based on a multi-output extension of the Cadzow denoising/annihilating filter method.