The stabilizability of a linear dynamical system (LDS) refers to the existence of control inputs that drive the system state to zero. In this article, we analyze both the theoretical and algorithmic aspects of the stabilizability of an LDS using sparse control inputs with potentially time-varying supports. We show that an LDS is stabilizable using sparse control inputs if and only if it is stabilizable (using unconstrained inputs). For a stabilizable LDS, we present an algorithm to determine the sparse control inputs that steer the system state to zero. We show that all stabilizable LDSs are also sparse mean square stabilizable when the process noise has zero mean and bounded second moment. For such an LDS, we devise a method to sequentially estimate the sparse control inputs to stabilize the LDS in the mean square sense. We prove that a detectable and stabilizable LDS is sparse stabilizable through output feedback and develop an algorithm for finding the corresponding sparse control inputs. Finally, we analyze the stabilizability of an LDS using sparse control inputs with common support. Our results shed light on the conditions under which a given LDS is stabilizable using sparse control inputs and the design of the corresponding control inputs.