  A continuous map $\mathbb{C}^n\to Gr(\tau, N)$ is $k$-regular if the
$\tau$-dimensional subspaces corresponding to images of any $k$ distinct points
span a $\tau k$-dimensional space. For $\tau = 1$ this essentially recovers the
classical notion of a $k$-regular map $\mathbb{C}^n\to \mathbb{C}^N$. We
provide new examples of $k$-regular maps, both in the classical setting $\tau =
1$ and for $\tau\geq 2$, where these are the first examples known. Our methods
come from algebraic geometry, following and generalizing
Buczy\'{n}ski-Januszkiewicz-Jelisiejew-Micha{\l}ek. The key and highly
nontrivial part of the argument is proving that certain loci of the Hilbert
scheme of points have expected dimension. As an important side result, we prove
irreducibility of the punctual Hilbert scheme of $k$ points on a threefold, for
$k\leq 11$.
