2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/129317We give a complete enumeration of all combinatorial 3-manifolds with 10 vertices: There are precisely 247882 triangulated 3-spheres with 10 vertices as well as 518 vertex-minimal triangulations of the sphere product $S^2\times S^1$ and 615 triangulations of the twisted sphere product $S^2_\times_S^1$. All the 3-spheres with up to 10 vertices are shellable, but there are 29 vertex-minimal non-shellable 3-balls with 9 vertices.9 pages, minor revisions, to appear in Beitr. Algebra GeomCombinatoricsGeometric Topology57Q15; 57N10text