There are multiple definitions of when points $P$ and $Q$ on schemes $X$ and $Y$ are equisingular. One can require that either $P$ and $Q$ have isomorphic analytic (i.e. complex analytic), or formal, or étale neighborhoods. My main question is how are these three definition related. I know that for simple singularities they are equivalent, and for difficult ones I guess they are non-equivalent. Does anyone know precise statements, references, proofs, examples when these are non-equivalent notions?


The analytic and the étale case is a little bit subtler. One can ask for the adequate local rings to be isomorphic and also for isomorphic analytic/étale neighborhoods, as above. Are these two approaches equivalent?



Analytic vs. formal vs. étale singularities
All these notions are equivalent; Mike Artin proved that. For the étale topology this is in "Algebraic approximation of structures over complete local rings", Inst. Hautes Études Sci. Publ. Math. No. 36 1969, 23–58; the analytic case it treated in "On the solutions of analytic equations", Invent. Math. 5 1968, 277–291. 


