  We say a closed point $x$ on a curve $C$ is sporadic if $C$ has only finitely
many closed points of degree at most $\operatorname{deg}(x)$ and that $x$ is
isolated if it is not in a family of effective degree $d$ divisors parametrized
by $\mathbb{P}^1$ or a positive rank abelian variety (see Section 4 for more
precise definitions and a proof that sporadic points are isolated). Motivated
by well-known classification problems concerning rational torsion of elliptic
curves, we study sporadic and isolated points on the modular curves $X_1(N)$.
In particular, we show that any non-cuspidal non-CM sporadic, respectively
isolated, point $x \in X_1(N)$ maps down to a sporadic, respectively isolated,
point on a modular curve $X_1(d)$, where $d$ is bounded by a constant depending
only on $j(x)$. Conditionally, we show that $d$ is bounded by a constant
depending only on the degree of $\mathbb{Q}(j(x))$, so in particular there are
only finitely many $j$-invariants of bounded degree that give rise to sporadic
or isolated points.
