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 $f : [0, 1]^d → R$, 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 $E_f (0)$.