How behave the typical $L^q$-dimensions of measures?

We compute, for a compact set $K\subset {\mathbb{R}}^d$, the value of the upper and of the lower $L^q$-dimension of a typical probability measure with support contained in $K$, for any $q\in \mathbb{R}$. Different definitions of the “dimension” of $K$ are involved to compute these values, following $q\in\mathbb R$.