Geoffrey Morley calls a tiling;

• Primitive if it has no triangle or 45-45-45-225 pseudotriangle subdivided into more than one or two triangles respectively. Otherwise the tiling is Derivative.
• Ultraperfect if no two triangles, subdivided or not, are the same size.
  
Morley has added thousands more perfects and has split them into separate catalogues of Isosceles Right Triangled Squares (IRTSs), namely;
• Primitive Perfects (PPIRTSs),
• Derivative Ultraperfects (DUIRTSs) and
• Non-Ultraperfect Perfects (NPIRTSs).
• Simple Primitive Imperfects (SPIIRTSs)

A fourth catalogue is for SPIIRTSs. A SPIIRTS is an imperfect isosceles right triangled square with no properly contained pseudotriangle/rectangle/triangle subdivided into 3/2/2 or more elements respectively. Two equal elements may constitute a non-square parallelogram but not a properly contained square.

Tilings of Rectangles can also be constructed. The attached pdf, by Geoffrey Morley, shows Simple Perfect Isosceles Right Triangled Rectangles (SPIRTRs) with aspect ratios of 1x2, 1x3, 1x4, 1x5 and 1x6.

A Pseudotriangle (Pseudoquadrangle) is a polygon with three (four) convex corners and an arbitrary number of reflex angles.
Precisely which orders of which catalogues are complete is not known.
Any letters which appear before order:side depict properties of the tiling:

• c = crossed. There is a tile-corner traversed by two lines. Example: PPIRTS 19:35AB1of4.
• d = double-pentagon patterned. Every such tiling is a subdivision of an instance of the same deformable tiling by two 45-90-90-90-225 pentagons with a shared side, four triangles and two pseudotriangles. Examples: PPIRTSs 13:26AA and 19:221AA, SPIIRTSs 8:4TA and 11:9TC. The only one of these examples in which none of the 8 underlying tiles is degenerate is PPIRTS 19:221AA. (A polygon is called degenerate when one or more sides have shrunk to zero length.)
• e = elegant. No tile-corner is just a T-junction. Such tilings may be considered aesthetically pleasing. Example: PPIRTS 13:21AA. There is no known example of an IRTS which is simultaneously crossed, elegant and perfect.
• f = as few as two elements in every subquadrilateral (or an SPIIRTS with no subquadrilateral).
• i = isomers exist which are ineligible for this catalogue. They are not included in the isomer count which follows 'of' in the tiling id.
• p/r/t = pseudotriangular/rectangular/triangular inclusion subdivided into at least 6/5/6 triangles respectively. Property 't' is omitted from NPIRTSs.
• z = zigzag by shorter sides of two or more equal tiles, pairs of which form parallelograms