There is considerable literature on this question, and closely related variations.
See:

- The
*Thompson problem*: Which configurations of $n$ electrons on a sphere minimize the electrostatic potential energy?
- The
*Tammes problem*: Which configurations of $n$ points on a sphere maximize the smallest distance between any two points? Sometimes phrased as packing $n$ congruent circles on a sphere:

^{
(Image from Paul Sutcliffe.)
}

According to

Musin, Oleg R., and Alexey S. Tarasov. "The Tammes problem for $N=14$." arXiv:1410.2536 Abstract (2014).

the Tammes problem is solved exactly for

- For $n=3,4,6,12$ by L. Fejes Toth (1943).
- For $n=5,7,8,9$ by Schütte and van der Waerden (1951).
- For $n=10,11$ by Danzer (1963).
**Added** (*8Sep15*): Exact radius for $n=10$ by Sugimoto & Tanemura.
- For $n=24$ by Robinson (1961).
- For $n=13, 14$ by Musin and Tarasov (2014).

^{
Fig.1 from Musin & Tarasov: $n=14$.
}

**Added** (*8Sep15*): The exact radius for $n=10$ was just found:

Teruhisa Sugimoto, Masaharu Tanemura.
"Exact value of Tammes problem for N=10." Sep 2015.
arXiv 1509.01768 Abstract.

^{
Fig.1b from Sugimoto & Tanemura.
}

**Added** (*31Dec2017*) in response to a question by @R_Berger:
For $n=20$, the best arrangement for the Tammes problem is not
the dodecahedron's vertices.
The optimal is unknown, but this beats the dodecahedron:

^{
Coordinates from Neil Sloane link,
due to R.H. Hardin, N.J.A. Sloane & W.D. Smith (1994).
}