Besides the many convex forms, which you can fill with DiDoms, there are some more interesting forms.

The Pythagoras triangle 3-4-5 is well known and the most famous. I found a form derived from this triangle in [MathPuzzle] Pythagoras. As the form is a bit too large, four Doms are added. I used (a bit different than Ed Pegg's original) another kite and two Doms. With four additional Doms there is a huge number of solutions.

The three kites naturally have always the same position.

There are 410 different solutions.

DiDom Pythagoras 1 (PDF).

The 5224 solutions of this form have different kite positions. The solution on the right has an interesting property, as you can draw a straight line through both grids.

DiDom Pythagoras 2 Sample (PDF)

The two ring constellations above are the only possible ways for the Pythagoras form to have Dom tilings. All other constellations result in parts where each one has an odd numbers of area units. Therefore solutions are only to be found with a modified form leaving two squares blank. This is just one example.

This form has no solution too, although both parts have an even number of units. There is no Dom tiling possible. You can under stand this, if you look at the chess colored version. The partial forms have a different number of black and white cells, where the Dom tiling needs equally many.

For real S-type symmetry all partial grids must have the same rotational center.

This form has 626 * 2 solutions. Only the two kites reside in the minimized base grid. For the same reason they have fixed positions.

The mirror symmetry axis at atan(½) divides the form into two equally shaped parts. The kites live on the mirror axis of course.

This form has 8 * 2 solutions.

Here we found 13 * 2 solutions with different kite positions.