With Alix Deruelle, we explain how Ricci flow develops or resolves 4-dimensional orbifold singularities by a notion of stability depending only on the curvature at the singular points. We construct a plethora of ancient and immortal Ricci flows spontaneously forming or desingularizing arbitrarily complicated orbifold singularities by bubbling-off Ricci-flat ALE metrics. This unexpectedly predicts that singular spherical and cylindrical orbifolds should not appear as finite-time singularity models. On the other hand, (complex) hyperbolic orbifolds appear as limits of immortal “thick” 4-dimensional Ricci flows as \(t\rightarrow +\infty\). With I. Lopez, we show that this is likely the only mechanism for orbifold singularity formation under a topological spin assumption.