Let $f:X\to Y$ be a universal homeomorphism of regular (excellent finite-dimensional) schemes, $Z\subset Y$ be a regular subscheme. Is $f^{-1}(Z)$ necessarily regular? 



Is the pre-image of a regular subscheme with respect to a universal homeomorphism of regular schemes regular?
Well, $f^{-1}Z$ could easily be non-reduced (for example, take the relative Frobenius morphism $\mathbb A^1\_k \to \mathbb A^1\_k$, defined by the embedding $k[y] = k[x^p] \subseteq k[x]$, where $k$ is a field of characteristic $p > 0$, and let $Z \subseteq \mathbb A^1$ be defined by $y = 0$), so I would guess that the question should be interpreted as asking whether $f^{-1}Z$ with its reduced structure is regular. But the answer is negative even in this case. For example, take the morphism $\mathbb A^2\_k \to \mathbb A^2\_k$ defined by the embedding $k[y,t] = k[x^p, t] \subseteq k[x, t]$, and let $Z \subseteq \mathbb A^2\_k $ be defined by $y + t^{p+1} = 0$.


