TITLE: Analytic vs. formal vs. étale singularities
QUESTION [16 upvotes]: 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?

REPLY [5 votes]: None of these notions is the usual definition of equisingular.  For example, any choice of four distinct lines though some given point in the plane is usually considered equisingular to any other, however, the analytic isomorphism class depends on the cross-ratio of the slopes.