The nature and properties of dark matter (DM) are both outstanding issues in physics. Besides clustering in halos, the universal character of gravity implies that self-gravitating compact DM configurations-predicted by various models-might be spread throughout the universe. Their astrophysical signature can be used to probe fundamental particle physics, or to test alternative descriptions of compact objects in active galactic nuclei. Here, we discuss the most promising dissection tool of such configurations: the inspiral of a compact stellar-size object and consequent gravitational-wave (GW) emission. The inward motion of this "test probe" encodes unique information about the nature of the supermassive configuration. When the probe travels through some compact region we show, within a Newtonian approximation, that the quasi-adiabatic inspiral is mainly driven by DM accretion and by dynamical friction, rather than by radiation reaction. When accretion dominates, the frequency and amplitude of the GW signal produced during the latest stages of the inspiral are nearly constant. In the exterior region we study a model in which the inspiral is driven by GW and scalar-wave emission, described at a fully relativistic level. Resonances in the energy flux appear whenever the orbital frequency matches the effective mass of the DM particle, corresponding to the excitation of the central object's quasinormal frequencies. Unexpectedly, these resonances can lead to large dephasing with respect to standard inspiral templates, to such an extent as to prevent detection with matched filtering techniques. We discuss some observational consequences of these effects for GW detection.