  We study rational points on a smooth variety X over a complete local field K
with algebraically closed residue field, and models of X with tame quotient
singularities. If a model of X is the quotient of a Galois action on a weak
N\'eron model of the base change of X to a tame Galois extension of K, then we
construct a canonical weak N\'eron model of X with a map to this model, and
examine its special fiber. As an application we get examples of singular models
of X such that X has K-rational points specializing to a singular point of this
model. Moreover we obtain formulas for the motivic Serre invariant and the
rational volume, and the existence of K-rational points on certain K-varieties
with potential good reduction.
