Many applications such as ultrasonic cleaning or sonochemistry use the ability of bubbles to oscillate and drive liquid flow. But bubbles have also received attention in porous media, where drying may cause cavitation, a phenomenon occurring in plant tissues. Here we explore the dynamics of cavitation bubbles when the liquid is fully entrapped in an elastic solid, using light scattering, laser strobe photography and high speed camera recordings. Our experiments show unexpectedly fast bubble oscillations in volume. They depend on the confinement size and elasticity, which we explain with a simple model where liquid compressibility is a key parameter. We also observe rich non-spherical dynamics, with ejection away from the walls and bubble fragmentation, which reveal extreme fluid motion at short timescales.