Suppose we want an integral model of the variety $X=\mathbb{A}^1_{\mathbb{Q}}$. Fix a prime $p$, and define $$ \tilde{X}=\mathrm{Spec}\big(\mathbb{Z}[x]\big)\,\backslash\,\big\{(p,x)\big\}. $$ Intuitively, the fiber of $\tilde{X}$ over $p$ looks different from all the other fibers. But I am having difficulty making this precise, as the variety $\tilde{X}$ is flat over $\mathbb{Z}$, and is even smooth.
In what way(s) can one formalize the idea that the integral model $\tilde{X}$ is "exceptional" at the prime $p$?