  We introduce the notion of a stratified Oka manifold and prove that such a
manifold $X$ is strongly dominable in the sense that for every $x\in X$, there
is a holomorphic map $f:\C^n\to X$, $n=\dim X$, such that $f(0)=x$ and $f$ is a
local biholomorphism at 0. We deduce that every Kummer surface is strongly
dominable. We determine which minimal compact complex surfaces of class VII are
Oka, assuming the global spherical shell conjecture. We deduce that the Oka
property and several weaker holomorphic flexibility properties are in general
not closed in families of compact complex manifolds. Finally, we consider the
behaviour of the Oka property under blowing up and blowing down.
