Squared squares and squared rectangles are called simple if they do not contain a smaller squared square or rectangle. Simple perfect squared rectangles begin at order 9.
Squared squares and squared rectangles are called perfect if the squares in the tiling are all of different sizes.
Simple perfect squared rectangles have been have been catalogued from order 9 to 17 in javascript menus. Initially arranged by order and then by width and height, but further selected catalogues arranged by other characteristics are planned.
The main method of classifying Squared Rectangles is to organise them with a given number of elements, by order, that is, the number of constituent squares, then by width then by height.
Catalogues have been produced of simple perfect squared rectangles(SPSR) in the search for squared squares.
The On-Line Encyclopedia of Integer Sequences lists sequence A002839; the number of perfect squared rectangles of order n as; 0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 22, 67, 213, 744, 2609, 9016, 31426, 110381. These are Duijvestijn's counts up to and including order 18. It is not practical to display catalogues above order 17 due to the sheer number of tilings involved.