We use a hybrid Monte Carlo algorithm to simulate the shaking of spheres at different vibrational amplitudes and find that spontaneous crystallization occurs in specific dynamical regimes. Several crystallizing transitions are typically observed, leading to end states which can be fully or partially ordered, depending on the shaking amplitude, which we investigate using metrics of global and local order. At the lowest amplitudes, crystallization is incomplete, at least for our times of observation. For amplitude ranges where crystallization is complete, there is typically a competition between hcp and fcc ordering. It is seen that fcc ordering typically predominates; in fact for an optimal range of amplitudes, spontaneous crystallization into a pure fcc state is observed. An interesting feature is the breakdown of global order when there is juxtaposition of fully developed hcp and fcc order locally: we suggest that this is due to the interfaces between the different domains of order, which play the same role as dislocations.