  In this paper we investigate two stratifications of the moduli space of
elliptically fibred K3 surfaces. The first comes from Shimada's classification
of connected components of elliptically fibred K3 surfaces and is closely
related to the root lattice of the fibration. The second is the monodromy
stratification defined by Bogomolov, Petrov and Tschinkel. The main result of
the paper is a classification of all positive-dimensional ambi-typical strata,
that is strata which are both Shimada root strata and monodromy strata. We
further discuss the connection with moduli spaces of lattice-polarised K3
surfaces. The paper contains an appendix by M. Kirschmer providing
computational results on the 1-dimensional ambi-typical strata.
