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The notions of shyness and prevalence generalize the property of being zeroand full Haar measure to arbitrary (not necessarily locally compact) Polishgroups. The main goal of the paper is to answer the following question: Whatcan we say about the Hausdorff and packing dimension of the fibers of prevalentcontinuous maps?  Let $K$ be an uncountable compact metric space. We prove that the prevalent$f\in C(K,\mathbb{R}^d)$ has many fibers with almost maximal Hausdorffdimension. This generalizes a theorem of Dougherty and yields that theprevalent $f\in C(K,\mathbb{R}^d)$ has graph of maximal Hausdorff dimension,generalizing a result of Bayart and Heurteaux. We obtain similar results forthe packing dimension.  We show that for the prevalent $f\in C([0,1]^m,\mathbb{R}^d)$ the set of$y\in f([0,1]^m)$ for which $\dim_H f^{-1}(y)=m$ contains a dense open sethaving full measure with respect to the occupation measure $\lambda^m \circf^{-1}$, where $\dim_H$ and $\lambda^m$ denote the Hausdorff dimension and the$m$-dimensional Lebesgue measure, respectively. We also prove an analogousresult when $[0,1]^m$ is replaced by any self-similar set satisfying the openset condition.  We cannot replace the occupation measure with Lebesgue measure in the abovestatement: We show that the functions $f\in C[0,1]$ for which positively manylevel sets are singletons form a non-shy set in $C[0,1]$. In order to do so, wegeneralize a theorem of Antunovi\'c, Burdzy, Peres and Ruscher. As acomplementary result we prove that the functions $f\in C[0,1]$ for which$\dim_H f^{-1}(y)=1$ for all $y\in (\min f,\max f)$ form a non-shy set in$C[0,1]$.  We also prove sharper results in which large Hausdorff dimension is replacedby positive measure with respect to generalized Hausdorff measures, whichanswers a problem of Fraser and Hyde.

Hausdorff and packing dimension of fibers and graphs of prevalent continuous maps