  We construct examples of spherical space forms $(S^3/\Gamma,g)$ with positive
scalar curvature and containing no stable embedded minimal surfaces, such that
the following happens along the Ricci flow starting at $(S^3/\Gamma,g)$: a
stable embedded minimal two-sphere appears and a non-trivial singularity
occurs. We also give in dimension $3$ a general contruction of Type I
neckpinching and clarify the relationship between stable spheres and
non-trivial Type I singularities of the Ricci flow. Some symmetry assumptions
prevent the appearance of stable spheres, and this has consequences on the
types of singularities which can occur for metrics with these symmetries.
