A set $A$ is nowhere dense if the interior of its closure is empty. $\mathbb{Z}$ is closed and its interior is empty (no open interval is contained in $\mathbb{Z}$), so $\overline{\mathbb{Z}}^0 = \mathbb{Z}^0 = \emptyset$. Thus $\mathbb{Z}$ is nowhere dense. $\mathbb{Q}$ is dense in $\mathbb{R}$ (not nowhere dense); $\mathbb{R}\setminus\mathbb{Z}$ is open and dense.