The recent paper On the Erdos distinct distance problem in the plane Authors: Larry Guth, Nets Hawk Katz prodded me to get a non-trivial example. Here is what I cannot find: an example of 12 distinct points in the plane with only 5 different distances between points. The regular 12-polygon has 6 different lengths but I cannot do better. http://oeis.org/A186704 implies that there is one>
Here goes my poor explanation:
Take a regular hexagonal lattice with distance 1 between nearest neighbors, and choose a 15-point equilateral triangle in this lattice (15 is a triangular number). Remove the 3 vertices of the triangle. You'll be left with 12 points and 5 distinct distances.
Edit: Just checked the OEIS reference, and it's available on Google Books. The picture you want is on page 200.
Thought it might be nice to show the set placed on the hexagonal lattice.
Here’s a Desmos link if you wish to play around with such point configurations.