Étale neighbourhoods and the normal crossings locus

Given a property of the complete local ring of a variety at a point, how can we show that the set of all points on the variety sharing the same property is open or closed in the Zariski topology? Such questions are ubiquitous especially in resolution of singularities. In the example of the normal crossings property and some variations we give a solution via étale neighbourhoods and Artin approximation.