A famous theorem of Carleson says that, given any function $f\in L^p(\TT)$,
$p\in(1,+\infty)$, its Fourier series $(S_nf(x))$ converges for almost every
$x\in \mathbb T$. Beside this property, the series may diverge at some point,
without exceeding $O(n^{1/p})$. We define the divergence index at $x$ as the
infimum of the positive real numbers $\beta$ such that $S_nf(x)=O(n^\beta)$ and
we are interested in the size of the exceptional sets $E_\beta$, namely the
sets of $x\in\mathbb T$ with divergence index equal to $\beta$. We show that
quasi-all functions in $L^p(\TT)$ have a multifractal behavior with respect to
this definition. Precisely, for quasi-all functions in $L^p(\mathbb T)$, for
all $\beta\in[0,1/p]$, $E_\beta$ has Hausdorff dimension equal to $1-\beta p$.
We also investigate the same problem in $\mathcal C(\mathbb T)$, replacing
polynomial divergence by logarithmic divergence. In this context, the results
that we get on the size of the exceptional sets are rather surprizing.