  In this paper, we prove that all irregular cusps on $X_1(N)$ of genus $\geq2$
are Weierstrass points except for $X_1(18)$. Also, for any positive integer $N$
of the form $p^2M$ with a prime $p$ and a positive integer $M$, we obtain some
results for when the irregular cusps of $X_0(N)$ equivalent to
$\left(\begin{smallmatrix}1\\p\end{smallmatrix}\right)$ are Weierstrass points
or not.
