By "regular", I'm going by a property of Delaunay Triangulation, which is to maximize the minimum angle. In $\mathbb{R}^2$, tiling with equilateral triangles gives you a minimum angle of 60 degrees. Any other triangulation would give you a smaller minimum angle, so in this sense, triangulating $\mathbb{R}^2$ with equilateral triangles is the most regular way.

What's the best way to triangulate $\mathbb{R}^3$ so that the minimum solid angle of the tetrahedra is maximized?

By "regular", I'm going by a property of Delaunay Triangulation, which is to maximize the minimum angle. In $\mathbb{R}^2$, tiling with equilateral triangles gives you a minimum angle of 60 degrees. Any other triangulation would give you a smaller minimum angle, so in this sense, triangulating $\mathbb{R}^2$ with equilateral triangles is the most regular way.

What's the best way to triangulate $\mathbb{R}^3$ so that the minimum solid angle of the tetrahedra is maximized?

I assume that the dual tessellation of closely-packed spheres would give a good triangulation since that works in $\mathbb{R}^2$. You get some regular tetrahedra and regular octahedra which can be divided into 4 congruent tetrahedra which contain the minimum solid angle. Is there a way to check if this is maximized?