Oscillating compact stars are promising sources of gravitational radiation. Upper limits on gravitational-wave (GW) burst emission associated with a Vela timing glitch and in coincidence with electromagnetic triggers from magnetars were set recently by the LIGO and Virgo Collaborations. Those searches targeted f-modes of neutron stars with standard equations of state. Indeed, the f-modes are believed to be the principal GW emitters of conventional neutron stars; their high frequencies, though, imply that we would be able to detect only very energetic nearby events. Exotic equations of state, on the other hand, which predict elastic quark matter cores, may be much more interesting. LIGO and Virgo continuous-wave searches are already able to put constraints on the physics of such objects by comparing theoretical models with data analysis (DA) results. In this presentation, we consider this scenario from the GW burst (and not the continuous-wave) point of view. We review the properties of oscillation modes of exotic compact stars with elastic cores. We make estimates on the GW energy output after a corequake and we discuss if/how we can use the latest burst DA results to extract information about the deep interiors of compact stars and/or the underlying physical mechanisms producing glitches and/or flares.