3½-ominoes?

Not strictly polyominoes but close enough really. I have no idea how I first found out about this set of pieces - I always had a hunch it was the set used in Martin Watson's puzzle 'DemiTri' (which doesn't look like it's on the site any more) but that says 12 pieces. And attempts to create similar sets in Peter Esser's program by slicing tetrominoes or adding half-squares to triominoes yields 12 and 13 piece sets respectively.

Fig. 1: The set, crudely rendered in Microsoft Paint.

But this looks like a complete set to me, all the ways of putting together three squares and a triangular half-square (if there's a fifteenth one and I've missed it let me know) and it's got a total area of 14 x 3.5 = 49 unit squares, which suggests (among other things) a 7x7 square:

Fig. 2: Here's one I made earlier.

Technique for solving these is a tad unusual. Since they each have one diagonal side, if the outer perimeter of the shape you're filling has no diagonal sides than the pieces are effectively 'paired up' by joining two at the diagonal edge. This results in any shape like this being split into seven heptominoes which can be in turn split in half to give two pieces. So my technique was to first put together a couple of promising looking heptominoes (i.e. ones containing 2x2 or 2x3 rectangles) then trying to fit those together. I used this method to cobble together the shapes below.

Fig 3. 10x5 with a bite taken out of it.

Fig. 4: These. Which can be put together to make the shape in Fig. 3.

Fig. 5: More shapes!

...and this nightmare shape that I found with a solver because there's no way I'd have the patience to do it by hand.

Sadly, these seem to be more limited with what you can do with them compared to, say, pentominoes or hexiamonds, both of which have a similar number of pieces (that, or I'm just really uncreative. I have a hunch it may be the latter.)
And there's also a scary bonus thought - this set of pieces is just one in a family. There's scope for doing things with the sets of pieces which are four squares and a triangle*, or two squares and two triangles, and so on, and at that point we're approaching just regular sets of polyaboloes or polytans or whatever people generally call them.
But that's going to have to be a post for another time.

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* If I counted correctly there's an odd number of these which might further limit what can be done with them.

20x20 Diagonal Square with Heptominoes

It turns out the heptominoes can do four congruent right-angled triangles, each with a single-square hole at the right angle:

Fig. 1: Four 19x19 triangles

I solved the first three quadrants of this by hand (the red, blue and green ones) then utterly despaired at the thought of having to do the fourth one. As a result of using my normal solving technique, I'd found myself left with a selection of mostly quite blocky, squarish pieces then realised that these pieces don't generally lend themselves well to building wiggly edges. For finishing off relatively square shapes like the last corner of a rectangle or something they're fine, but for this I wasn't sure. So I wrote it off as a bad job.
Then a few days later, I drew the pieces into FlatPoly2 just for the sheer hell of it and it found a solution in about a minute.

These four triangles can then be put together in various different ways, including this 19x19 (Edit: it's 20x20, I can't count) diagonal square:

Fig. 2: The holes don't quite match the symmetry of the outer perimeter but whatever.

'Measure twice, construct once'

Building something only to discover that I've somehow put the central holes in the wrong place seems to be a common theme for me. I need to implement some 'measure twice, construct once' philosophy maybe.
A short while after I wrote that big long post about combining pentominoes and hexominoes into one set, I thought about how just possible it would be to knock out a 4xn rectangle with those pieces. I mean, each piece would physically fit into a 4 cell high rectangle, and if it's possible to squeeze all 1- to 7-ominoes into a 5x211* then this could well be possible too.

The whole 4-cell-high thing made it tricky but not as big a challenge as I had initially suspected it was going to be. After 45 minutes with my hexomino set (and a shoddy set of pentominoes cut out of graph paper to supplement them) I found the pattern below. With the two holes off-centre by one...

Fig. 1a: Side note, I'm not mad on this arrangement of holes but it's the most symmetrical one I can think of for two holes in an even x even rectangle.

Thankfully, as I was sketching down this solution, I notices it wasn't as bad as I'd feared - there was a central section (marked in light gray above) which could be flipped over and had the effect of transposing the two dots over by one cell - into the middle!

Fig. 1b: The finished construction.

Of course, there's rarely a nice quick fix like that, as I discovered with a heptomino construction from a few weeks ago. The harbour heptomino and its little central hole must have been accidentally knocked during the solve process and I hadn't realised. Only upon drawing the solution down onto graph paper did I notice it was slightly off. There was no quick little rearrangement of pieces that would fix it this time. I just had to solve the entire thing again in a few days' time.

Fig. 2: The heptominoes in an octagon with central hole. Coloured by how far each heptomino is from the edge of the construction, just to see how it would look really.

Notice the way the diagonally-symmetric pieces are all grouped nicely along a diagonal running from near the south-west edge up through the centre. Basically, I noticed in one of David Bird's nonomino constructions (typically, the one I can't find on the internet anywhere to link to; it's in S.W. Golomb's 'Polyominoes', page 116) that all (or at least a good chunk) of the rotationally symmetric pieces were grouped together in the central section surrounded by holes**. And somehow I'd never noticed that before. I just like the idea of little things like that, 'hidden' in plain sight within the construction that reveal themselves on closer inspection. (Another one I like is in a heptomino construction by Nick Maeder, third one from the bottom on this page, which has the crucifix-shaped heptomino in the centre of a triplicated version of itself, all positioned centrally within the pattern.)

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* I've seen this done but can't find it anywhere on the internet, but can't find in anywhere or I'd link it. Maybe it was in the middle of an Internet Archive binge of polyomino-related sites that are no longer up.

** Actually, there's something similar going on in this nonomino construction, also by David Bird. A lot of symmetrical pieces are grouped around the line of holes running upwards from the middle.