The classical convergence theory and singularity formation for non-collapsed Einstein 4-manifolds leaves some important issues open. Which singular Einstein metrics are actually limits of smooth ones? Can we describe the moduli space of Einstein metrics close to its boundary? We partially answer these by proving that any Einstein manifold sufficiently GH-close to an Einstein orbifold is the result of a gluing-perturbation procedure. This completely describes the singularity formation of compact Einstein metrics and moreover lets us show that the desingularization of Einstein metrics by Kronheimer's gravitational instantons is obstructed. The recent results of Biquard-Hein let us extend a weaker obstruction for general Ricci-flat ALE spaces.