Certain classes of solutions to Einstein’s field equations admit singularities from which light can escape, known as ‘naked5 singularities Such solutions contradict the Cosmic Censorship hypothesis, however they tend to occur in spacetimes with a high degree of symmetry. It is thought that naked singularities are artifacts of these symmetries, and would not survive when the symmetry is broken. In particular, a rich source of naked singularities is the class of self-similar spherically symmetric spacetimes. It is the purpose of this thesis to test the stability of these solutions and examine if the naked singularity persists. We first consider the propagation of a scalar field on these background spacetimes and then study gauge-invariant perturbations of the metric and matter tensors. We exploit the spherical symmetry of the background to decompose the angular part of the perturbation in terms of spherical harmonics. Then we perform a Mellin transform of the field to reduce the problem to a set of coupled ordinary differential equations, and seek solutions for the individual modes. The asymptotic behaviour of these modes near singular points of the ODE’s is used to calculate a set of gauge invariant scalars, and we examine the finiteness of these scalars on the naked singularity’s horizons. The background spacetimes we examine are the self-similar null dust (Vaidya) solution, the self-similar timelike dust (Lemaitre-Tolman-Bondi) solution, and finally a general self-similar spacetime whose matter content is unspecified save for satisfying the dominant energy condition. In each case examined we find the Cauchy horizon, signalling the presence of a naked singularity, is stable to linear order, a surprising result that suggests naked singularities may arise in physical models of gravitational collapse. The second future similarity horizon which follows the Cauchy horizon is unstable, which suggests that the naked singularity is only visible for a finite time.