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. 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.

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. 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.