  Let $k$ be an arbitrary field. We construct examples of regular local
$k$-algebras $R$ (of positive dimension) for which the ring of differential
operators $D_k(R)$ is trivial in the sense that it contains {\it no} operators
of positive order. The examples are excellent in characteristic zero but not in
positive characteristic. These rings can be viewed as being non-singular but
they are not simple as $D$-modules, laying to rest speculation that
$D$-simplicity might characterize a nice class of singularities in general. In
prime characteristic, the construction also provides examples of {\it regular}
local rings $R$ (with fraction field a function field) whose Frobenius
push-forward $F_*^eR$ is {\it indecomposable} as an $R$-module for all $e\in
\mathbb N$. Along the way, we investigate hypotheses on a local ring $(R, m)$
under which $D$-simplicity for $R$ is equivalent to $D$-simplicity for its
$m$-adic completion, and give examples of rings for which the differential
operators do not behave well under completion. We also generalize a
characterization of $D$-simplicity due to Jeffries in the $\mathbb N$-graded
case: for a Noetherian local $k$-algebra $(R, m, k)$, $D$-simplicity of $R$ is
equivalent to surjectivity of the natural map $D_k(R)\to D_k(R, k)$.
