I've got an update to z4k's answer. It's possible it's because of changes since he posted his answer eight years ago.

First off, I'd recommend including sizes 36 and 72 because they are used for program icons in the start menu and for medium desktop icons respectively at the popular 144 pixels per logical inch / 150% scaling factor.

Secondly, I'd recommend excluding sizes 31 and 47, because they're only used for small start menu tiles and shown at the wrong sizes, 30 and 42 respectively, so they'll be blurry anyway. (This is actually a common theme. I've noticed a lot of blurry icons everywhere on recent Windows versions so I can only assume that Microsoft's programmers have become less competent over the years.) I also recommend excluding 63 which may either have been a copying error on z4k's part or the result of a really obscure bug, I cannot image.

The final list of recommended resolutions then becomes: 16, 20, 24, 28, 30, 32, 36, 40, 42, 48, 56, 60, 72, 84, 256

If your program has to run on older systems or work better with remote desktop you also have to include low colour icons. I'd recommend the following: 32 monochrome if you need it, and 16, 32 and 48 in both 16 colours and 256 colours. As a matter of fact at the small size of an icon the difference between true colour and 256 colours may not be possible to spot and it keeps the file size down as well so you may opt to also use that for larger sizes. As a final note, some programming frameworks have the annoying tendency to be picky about the order in which the icons appear in the file. I vaguely remember having once used one where it would use the wrong icon unless size 16 was first in the file. So if you don't see the right icon, experiment with the order of the sizes in the file.