If $\mathbb{C}[G] \hookrightarrow \mathbb{C}[H]$ is an extensionof Hopf domains of degree $d$, then $H \twoheadrightarrow G$is an \'etale map. Equivalently, the variety$X_{\mathbb{C}[H]}$ of $d$-dimensional $\mathbb{C}[H]$-modulescompatible with the trace map of the extension, is a smooth$\mbox{GL}_d$-variety with quotient $G$.If we replace $\mathbb{C}[H]$ by a non-commutative Hopfalgebra $H$, we construct similarly a $\mbox{GL}_d$-varietyand quotient map $\pi : X_H \twoheadrightarrow G$. Thesmooth locus of $H$ over $\mathbb{C}[G]$ is the set ofpoints $g \in G$ such that $X_H$ is smooth along$\pi^{-1}(g)$.We relate this set to the separability locus of $H$ over$\mathbb{C}[G]$ as well as to the (ordinary) smoothlocus of the commutative extension$\mathbb{C}[G] \hookrightarrow Z$ where $Z$ is the centre of $H$.In particular, we prove that the smooth locus coincideswith the separability locus whenever $H$ is a reflexiveAzumaya algebra. This implies that the quantum functionalgebras $O_{\epsilon}(G)$ and quantised enveloping algebras$U_{\epsilon}(\mathfrak{g})$ are as singular as possible. 1991 Mathematics Subject Classification: 16W30, 16R30.