I like to think that compact 3-manifolds will not be known until we know "the list" of compact, oriented, hyperbolic 3-manifolds, in the way that we know "the list" of compact, oriented, hyperbolic surfaces: the genus 2 surface, the genus 3 surface, the genus 4 surface, ...

We don't know "the list" yet, as the other answers indicate. But the potential for ordering hyperbolic 3-manifolds by volume was established by Jorgensen who proved that the set of volumes of finite volume hyperbolic 3-manifolds is a well-ordered set with order type $\omega^\omega$ (the same order type as polynomials with natural number coefficients). And with that in mind, we know the first entry: the Weeks manifold, which is the unique lowest volume closed hyperbolic 3-manifold, as proved by Gabai, Meyerhoff, and Milley. I'm not up on the very latest developments of this technology, but I think that the next few lowest volumes have also been completely listed, using the MOM technology of the same three authors.