A counterexample is to let $X_t$ be Brownian motion with drift. Start at any point $x$ and suppose the drift is negative.
Let $N_y$ be the event that $y$ is never hit, i.e., $N_y=\{(\forall t)\, X_t < y\}$.
With probability one there will be some positive value that is not hit; see e.g. [this question][1].
So
$$
\mathbb P (\cup_{y\in\mathbb N}\, N_y) = 1.
$$
Therefore
$$
\exists y\in\mathbb N\qquad \mathbb P(N_y)>0,
$$
and such a $y$ is a counterexample to regularity.


  [1]: https://math.stackexchange.com/questions/169548/max-of-brownian-motion-with-drift-is-finite-almost-surely