Choosing what type of sphere to use was no easy task. I needed a sphere that had an even distribution of detail and a high level of control over the resolution of the mesh.
The default option is an UV sphere, which is evenly divided into rows and columns. This allows for a high level of granularity of control over the mesh's detail, which is one half of the equation. However, this type of sphere has the problem of having a higher density of vertices and faces near the poles of the sphere. This would not work well for my purposes as I don't want the planet to have higher levels of details near the poles.
The second option would be to use an ico sphere, which has equally sized triangles. This gives me an even distribution of detail across the sphere. However, due to the nature of equally subdividing triangles, I can only increase the number of faces by a factor of four. This gives me very little granularity and control of how detailed the sphere is.
The third option is to start with a cube, and then normalize each vertices position from the center. This creates a sphere has a very even distribution of detail, with only slightly higher level of detail near the seams where the sides of the cube have come together. It also gives me a high granularity of detail as I can divide the sides of the cube into as many faces as I'd like. The only downside is that there can be some small artifacts along the seam lines where the sides of the cube meet.
Using a normalized sphere has turned out great. It satisfied all of my requirements and the math needed to create it was fairly simple. The small artifacts along the seam lines, where some vertices just don't quite connect, are all but invisible once noise is applied and the sphere is deformed. I did find a solution that solves this problem, which is to merge vertices that are really close to each other, connecting the small gaps. However, merging nearby vertices takes about 500 milliseconds of compute time on my powerful desktop computer. This performance hit is certainly not worth the small benefit of eliminating a rare and difficult to spot artifact.