One way around this is cube-mapping. You construct a cube then normalize all the vertices to "over-inflate" it until it's a sphere. Then you have six textures mapped to the faces of the cube, and no cat's bumhole, and no international date-line zipper. If you subdivide the faces of the cube into rectangles via equal angles instead of naively into a grid, then cut each of the rectangles into two triangles by the shortest diagonal, then you get a very nice tessellation with triangles that are nice and fat (nearly equilateral) with no thin sliver triangles and they're all of approximately equal size. The other thing you can do is double the vertices along the cube face edges, so that no two cube faces share any vertices. In that way you can avoid some "hairy ball theorem" related problems when it comes time to do normal mapping, which involves constructing tangent and bitangent vectors for each vertex. Each face of the cube can have a field of tangents and bitangents with no discontinuities, and since no texture or normal map crosses any boundary between faces, you avoid problems that come up with such discontinuities. (Hairy ball theorem says it's impossible to construct a field of tangents and bitangents covering a sphere that does not contain some discontinuity, so one solution is to stuff the discontinuities between cube map faces where they cannot bother any texture or normal map.)

Tessellated sphere looks like this: https://imgur.com/l3GmWq3