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Jim's example is good; one can easily construct examples by using elliptic fibrations. Let $$S:=X\times E\rightarrow X$$ be a trivial elliptic fibration with $E$ an elliptic curve. Construct a branched cover $Y\rightarrow X$. The base change of $S\rightarrow X$ will be a surface $$S'\rightarrow Y$$ with multiple fibers over the branch points of $Y\rightarrow X$. The difference of the set-theoretic fibers of two branch points with multiplicity $m$ is $m$-torsion in homology.