We propose a framework of stability analysis for a class of linear non-autonomous hybrid systems, with solutions evolving in continuous time governed by an ordinary differential equation and undergoing instantaneous changes governed by a difference equation. Furthermore, the jumps may also be triggered by exogeneous hybrid signals. The proposed framework builds upon a generalization of notions of persistency of excitation (PE) and uniform observability (UO), which we redefine to fit the realm of hybrid systems. Most remarkably we propose for the first time in the literature a definition of hybrid persistency of excitation. Then, we establish conditions, under which, hybrid PE implies hybrid UO and, in turn, uniform exponential stability (UES) and input-to-state stability (ISS). Our proofs rely on an original statement for hybrid systems, expressed in terms of Lp bounds on the solutions. We also demonstrate the utility of our results on generic adaptive estimation problems. The first one concerns the so-called gradient systems, reminiscent of the popular gradient-descent algorithm. The second one pertains to the design of adaptive observers/identifiers for a class of hybrid systems that are nonlinear in the input and in the output, and linear in the unknown parameters. In both cases, we illustrate through meaningful examples that the proposed hybrid framework succeeds in scenarii where the classical purely continuous or discrete-time counterparts fail.