There are clearly an infinite number of solutions, just as there are to the two body problem. My take is that they are finding new classes of solutions.
One example of the three body problem is the moon-earth-sun. The system is powerfully perturbed by Jupiter and Venus, so it’s not really a pure three body problem, but it is representative of one class of solutions: Ms >> Me >> Mm, where “>>” means “is sufficiently greater than” with the two smaller bodies body in a nearly Keplerian orbit, and their center of mass orbiting the most massive central body . Another class are the famous Lagrangian points, which include thousands of Trojan (no smirking) asteroids. Another is the horseshoe (only in Lagrangian coordinates) orbit. It was believed that horseshoe orbits were only possible theoretical constructs, until asteroids in horseshoe orbits around earth were discovered.
Holy cow, my jaw dropped when I saw the MOTH III class. Maybe some science fiction writer of the future will set his novel in such a system? I hope so.
Poincaré... Way before my time and way over my head, lol!