  Let $X_\Delta(N)$ be an intermediate modular curve of level $N$, meaning that
there exist (possibly trivial) morphisms $X_1(N)\rightarrow X_\Delta(N)
\rightarrow X_0(N)$. For all such intermediate modular curves, we give an
explicit description of all primes $p$ such that
$X_\Delta(N)_{\overline{\mathbb F}_p}$ is either hyperelliptic or trigonal.
Furthermore we also determine all primes $p$ such that $X_\Delta(N)_{\mathbb
F_p}$ is trigonal.
  This is done by first using the Castelnuovo-Severi inequality to establish a
bound $N_0$ such that if $X_0(N)_{{\overline{\mathbb F}_p}}$ is hyperelliptic
or trigonal, then $N \leq N_0$. To deal with the remaining small values of $N$,
we develop a method based on the careful study of the canonical ideal to
determine, for a fixed curve $X_\Delta(N)$, all the primes $p$ such that the
$X_\Delta(N)_{ {\overline{\mathbb F}_p}}$ is trigonal or hyperelliptic.
  Furthermore, using similar methods, we show that
$X_\Delta(N)_{{\overline{\mathbb F}_p}}$ is not a smooth plane quintic, for any
$N$ and any $p$.
