  The notions of shyness and prevalence generalize the property of being zero
and full Haar measure to arbitrary (not necessarily locally compact) Polish
groups. The main goal of the paper is to answer the following question: What
can we say about the Hausdorff and packing dimension of the fibers of prevalent
continuous 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 Hausdorff
dimension. This generalizes a theorem of Dougherty and yields that the
prevalent $f\in C(K,\mathbb{R}^d)$ has graph of maximal Hausdorff dimension,
generalizing a result of Bayart and Heurteaux. We obtain similar results for
the 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 set
having full measure with respect to the occupation measure $\lambda^m \circ
f^{-1}$, where $\dim_H$ and $\lambda^m$ denote the Hausdorff dimension and the
$m$-dimensional Lebesgue measure, respectively. We also prove an analogous
result when $[0,1]^m$ is replaced by any self-similar set satisfying the open
set condition.
  We cannot replace the occupation measure with Lebesgue measure in the above
statement: We show that the functions $f\in C[0,1]$ for which positively many
level sets are singletons form a non-shy set in $C[0,1]$. In order to do so, we
generalize a theorem of Antunovi\'c, Burdzy, Peres and Ruscher. As a
complementary 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 replaced
by positive measure with respect to generalized Hausdorff measures, which
answers a problem of Fraser and Hyde.
