Structural freeplay due to loosened mechanical linkages is a discrete nonlinear event which occurs pseudo-routinely in modern aircraft, causing severe airframe vibration. This impacts fatigue life, and has serious implications for fleet management and Structural Health Monitoring (SHM). While the concepts which drive SHM for aircraft are traditionally based on reactive procedures, we are currently observing a major shift towards actionable and pro-active condition-based maintenance, known as Prognostics and Health Management (PHM), to significantly reduce fleet sustainment costs. Given this current paradigm shift, there is a demand for the development of novel strategies to address decades old SHM problems, where the adaptation of existing methods or the development of new and innovative techniques both play critical roles. In this paper a signal processing framework is presented, based upon well-established nonlinear system identification methods, to rapidly diagnose structural freeplay in aircraft systems with a focus on the requirements of PHM technology. The framework exploits the nonlinear dynamical characteristics of the structural freeplay anomaly in a transonic aeroelastic system by specifically targeting rich bilinear signatures that are encoded in time-domain sensory outputs, via the Higher-Order Spectra (HOS) and the Empirical Mode Decomposition (EMD). The characteristic freeplay signatures which were initially extracted from computational transonic aeroelastic models are shown to be analogous in a transonic flight-test case-study (an all-movable horizontal tail with actuator freeplay), presenting a rare and important opportunity to verify the practical freeplay identification research. Once verified, a comprehensive understanding of the fundamental bilinear signatures allows the HOS and EMD to be adapted and refined towards a structured freeplay diagnosis framework. Using the extensive flight-test dataset as a case study, it is shown that the freeplay lo