  We study the singularity (multifractal) spectrum of continuous convex
functions defined on $[0,1]^{d}$. Let $E_f({h}) $ be the set of points at which
$f$ has a pointwise exponent equal to $h$. We first obtain general upper bounds
for the Hausdorff dimension of these sets $E_f(h)$, for all convex functions
$f$ and all $h\geq 0$. We prove that for typical/generic (in the sense of
Baire) continuous convex functions $f:[0,1]^{d}\to \mathbb{R} $, one has $\dim
E_f(h) =d-2+h$ for all $h\in[1,2],$ and in addition, we obtain that the set $
E_f({h} )$ is empty if $h\in (0,1)\cup (1,+\infty)$. Also, when $f$ is typical,
the boundary of $[0,1]^{d}$ belongs to $E_{f}({0})$.
