Hartshorne I.5 mentions the definition of being analytically isomorphic: P on X and Q on Y are analytically isomorphic iff the completion of O_P is isomorphic to the completion of O_Q where the completion is according to their maximal ideal in that local field.
For any pair of regular points, they are always analytically isomorphic as long as they are of the same dimension according to Cohen Structure Theorem.
But for singularities, they might be different.
My question is: Is classification of the completion of O_P a first step to classify all singularities(on curves in particular)? What are known results about that classification? What does that imply if the classification of rings after completion was complete? WHAT IS THE CORRECT WAY TO CLASSIFY SINGULARITIES?