Point sets in Euclidean space with a small number of distinct distances

Here I mention some asymptotic results—valid when the number of points $n$ grows large—which may not be directly relevant to your concentration on few distances.

The 2003 paper, "Distinct distances in three and higher dimensions," by Aronov, Pach, Sharir, Tardos, established that the number of distinct distances determined by $n$ points in $\mathbb{R}^3$ is $\Omega( n^{77/141 - \epsilon} )$ for any $\epsilon > 0$. Their result holds for points on a sphere as well. For $\mathbb{R}^d$ they achieved a lower bound of about $n^{1/(d-90/77)}$, again also for points on a $d$-sphere.

These lower bounds can be contrasted to the number of distinct distances achieved by points in a $n^{1/d} \times ... \times n^{1/d}$ integer lattice, which is $O(n^{2/d})$. Erdős conjectured the matching lower bound $\Omega(n^{2/d})$.

A bit later (2006), their results were improved by Solymosi and Vu in the paper, "Near optimal bounds for the Erdős distinct distances problem in high dimensions," establishing $\Omega(n^{(2/d)-2/(d(d+2))})$.

There's a nice book by Garibaldi, Iosevich, and Senger, The Erdős Distance Problem, in the Student Mathematical Library series of the American Mathematical Society (AMS link). Mostly it's about the problem in the plane, but there is some discussion of, and references to, work on higher dimensions.

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This paper:

http://maths.ucd.ie/~osburn/lattices.pdf

Has very cool results/connections of this (essentially, the question is: if you assume that lattices are best, which lattices are best among them).