  In this article we study combinatorial degenerations of minimal surfaces of
Kodaira dimension 0 over local fields, and in particular show that the `type'
of the degeneration can be read off from the monodromy operator acting on a
suitable cohomology group. This can be viewed as an arithmetic analogue of
results of Persson and Kulikov on degenerations of complex surfaces, and
extends various particular cases studied by Matsumoto, Liedtke/Matsumoto and
Hern\'andez-Mada. We also study `maximally unipotent' degenerations of
Calabi--Yau threefolds, following Koll\'ar/Xu, showing in this case that the
dual intersection graph is a 3-sphere.
