Let $N$ be an integer $\geq 3$ and $X(N)\rightarrow \mathrm{Spec } \mathbb{Z}[1/N]$ is the projective smooth modular curve defined in Deligne-Rappoport. Is there an exemple of $N$ for which the special fibre $X_{\mathbb{F}_p}(N)$ of $X(N)$ at some prime $p \nmid N$ is isomorphic to the projective $\mathbb{P}^{1}_{\mathbb{F}_p}$?
Another question : Is there some technics to compute the genus of $X_{\mathbb{F}_p}(N)$?
(1) The Riemann-Hurwitz applied to the covering $X(N)\to X(1)$ (over the complex numbers) furnishes a number theoretical formula for the genus of $X(N)$. (The degree of this covering is the cardinality of $\mathop{\rm SL}(2,\mathbf Z/N\mathbf Z)/\{\pm \mathrm I_2\}$, the ramification can be analysed.) Unless $N$ is small (ie, $N\leq 5$), the genus will be strictly positive. Formulas are to be found in probably all good textbooks about modular forms (but Wikipedia only gives the computation for prime number $N$).
(2) As you write, the scheme $X(N)$ is projective and smooth over $\mathop{\rm Spec}(\mathbf Z[1/N])$. In particular, the genus of all fibers are the same.
