Earthquake nucleation may begin with a stable quasi-static localization of fault slip. This long-term process, which conditions the fault for the rupture, is followed by a dynamic and unstable phase that we call initiation and ends with the onset of rupture propagation. In this paper, we investigate the three-dimensional elastodynamics of this unstable initiation phase both analytically and numerically. The fault model consists of two symmetric semi-infinite elastic bodies in frictional contact across a flat interface with a slip-weakening law and loaded until its sliding threshold is reached. On the basis of our previous studies in two dimensions, we generalize the linear stability and the spectral analysis and then solve analytically the dominant part approximation for the homogeneous infinite fault. We then derive the slip pattern on the fault, the decay of the displacement amplitude in the media, and its growth in time. The model also gives a theoretical approximation for the time of initiation. To test the dominant part approximation, we compare it with the numerical solution in the case of a concentrated perturbation. Finally, we find that unlike the infinite fault, finite faults of a given geometry allow a domain of stability that depends on the fault size. When a finite fault becomes unstable and rupture initiates, the slip growth is determined by a single dominant eigenmode. The duration of the initiation phase varies strongly with the size and geometry of the fault, especially when it is close to the stability limit.