Phase-field models, sometimes refered to as gradient damage or smeared crack models, are widely used methods for the numerical simulation of crack propagation in brittle materials. Theoretical results and numerical evidences show that they can predict the propagation of a pre-existing crack according to Griffith' criterion. For a one-dimensional problem, it has been shown that they can predict nucleation upon a critical stress, provided that regularization parameter be identified with the material's internal or characteristic length. In this article, we draw on numerical simulations to study crack nucleation in commonly encountered geometries for which closed-form solutions are not available. We use U-and V-notches to show that the nucleation load varies smoothly from that predicted by a strength criterion to that of a toughness criterion, when the strength of the stress concentration or singularity varies. We present validation and verifications numerical simulations for both types of geometries. We consider the problem of an elliptic cavity in an infinite or elongated domain to show that variational phase field models properly account for structural and material size effects. We conclude that variational phase-field models can accurately predict crack nucleation through energy minimization in a nonlinear damage model instead of introducing ad-hoc criteria.