Differences between singularity of total space and singularity of fibers

Question

Suppose in the category of varieties over a field $k$ of characteristic zero we have a fibration \begin{equation} \pi:X \rightarrow S \end{equation} Suppose further $S$ is a smooth. I know examples where $X$ is smooth, while some fibers of this family can become singular. In general, are there any kind of relations between the singularity of $X$ and the singularities of fibers?

Answer 1

Yes, there are often relations between the singularities of the fibers and the singularities of the total space. An early example of such a relation is that if $X_s$ has rational singularities for all $s$ and $\pi$ is flat, then $X$ has rational singularities. This was proven by Elkik in "Singularities rationelles et deformations", Inventiones 47 (1978). Similar theorems have been proven for other classes of singularities - if you google about "deformations of ADJECTIVE singularities", where ADJECTIVE is replaced with the class of singularities you care about, you will probably find good discussion or papers about this fact.