  Let $1 \leq d < D$ and $(p,q,s)$ satisfying $0 < p < \infty$, $0 < q \leq
\infty$, $0 < s-d/p < \infty$. In this article we study the global and local
regularity properties of traces, on affine subsets of $\R^D$, of functions
belonging to the Besov space $B^{s}_{p,q}(\R^D)$. Given a $d$-dimensional
subspace $H \subset \R^D$, for almost all functions in $B^{s}_{p,q}(\R^D)$ (in
the sense of prevalence), we are able to compute the singularity spectrum of
the traces $f_a$ of $f$ on affine subspaces of the form $a+H$, for
Lebesgue-almost every $a \in \R^{D-d}$. In particular, we prove that for
Lebesgue-almost every $a \in \R^{D-d}$, these traces $f_a$ are more regular
than what could be expected from standard trace theorems, and that $f_a$ enjoys
a multifractal behavior.
