A classic and fundamental result in number theory is due to Mordell who proved that the set of points on an elliptic curve defined over a number field forms a finitely generated abelian group; in particular, it has a finite torsion subgroup. An essential tool to study elliptic curves is the modular curves which are moduli spaces for elliptic curves with an additional structure. In particular, $X_1(n)$ classifies the elliptic curves with a point of order of $n$. Motivated by the classification of torsion problems, we study the sporadic points on the curve $X_1(n)$, that is, the closed points on $X_1(n)$ such that there are at most finitely many points of degree at most $\deg(x)$. In this talk, we will discuss the finiteness of sporadic points. This is joint with A. Bourdon, Y. Liu, F. Odumudu and B. Viray.