The recent paper On the Erdos distinct distance problem in the plane Authors: Larry Guth, Nets Hawk Katz prodded me to get a nontrivial 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 12polygon 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 15point 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.
If I may supplement Logan Maingi's apposite answer with a snapshot of the page to which he refers:
(I couldn't resist including the surrounding conjecture.)

$\begingroup$ Thanks! I tried to no avail to embed the image, but I'm still rather bad with TeX and couldn't get it to work. $\endgroup$ – Logan M Mar 14 '11 at 14:35

3$\begingroup$ Coming to this very late, but this suggests an interesting secondary conjecture: that a maximal set of points for $k$ distinct distances can always be realized as a subset of the triangular grid. Given the form of the conjecture I presume this has to be wrong, but is a specific $k$ known for which this isn't true? $\endgroup$ – Steven Stadnicki Mar 13 '17 at 2:41
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.