Given a hyperbolic manifold, the spectrum of its Laplacian contains a great deal of information about its geometric structure. In this talk, I will present joint work with Will Hide, Bram Petri, and Joe Thomas in which we study the spectral gaps -the first nonzero eigenvalue- of two models of random hyperbolic 3-orbifolds related to the Apollonian group. If time permits, I will also mention how this can be used to investigate the bass note spectrum of the set of hyperbolic 3-orbifolds.