Phonon cavities are believed to be the next step towards a control of dephasing in semiconductor quantum dot `qubits'. In this paper, we discuss two models for phonon cavities - a surface acoustic wave (SAW) inter-digitated transducer on an infinite half-space, and an elastic thin slab. The inelastic current through double quantum dots in non-perfect SAW cavities exhibits a gap at small energies and is completely suppressed in a perfect, infinite system. In the free-standing slab model, van Hove singularities evolve in the phonon spectral density. We find that these singularities cause additional side peaks in the inelastic current.