The modular curve X1(N) parametrizes elliptic curves (whose points form an abelian group under a geometric group law) with a marked point of order N. A point on X1(N) is isolated if, roughly speaking, it is not a member of an infinite family of points parametrized by a geometric object. Elliptic curves without CM naturally give isolated points on X1(N), but the non-CM points remain mysterious. We develop an algorithm to test whether an elliptic curve E/Q without complex multiplication gives rise to an isolated point of any degree on any modular curve of the form X1(N). Running this algorithm on all elliptic curves presently in the L-functions and Modular Forms Database and the Stein-Watkins Database gives strong evidence for our conjecture that only finitely many rational j-invariants give rise to an isolated point on X1(N) for any N; our conjectured list is explicit. This is joint work with Bourdon, Hashimoto, Keller, Klagsburn, Najman, and Shukla.