Singular Fibrations with ADE Singularities

Question

If we have a fibration (i.e. satsfying the homotopy lifting property) E $\rightarrow$ M with M a smooth complex manifold and E a potentially singular variety and the fibres having at worst ordinary double point singularities, what types of singular behaviors could we see in the total space E and under what circumstances would the total space E be smooth complex manifold. Specifically in the case I am thinking about the smooth fibres would be modeled on smooth K3-Surfaces and the singular fibres would be K3-Orbifolds. More generally if we have the same basic setup but the singular fibres have general ADE singularities what would be the implication for the total space, and are there conditions that one could put to ensure there are ADE singularities in the total space as well?