Where can I find load ratings for two-by lumber laid flat?
The reason you can't find tabulated information on dimension lumbar laid flat (usually called "weak axis bending") is because the strength is so small that there is virtually no structural application. The only time you would use 2x lumber like that is something like flooring where the boards are almost continuously supported and are carrying very little weight.

However if you want to get a sense of the strength you can compare the moment of inertia of the various cross-sections and orientations. Bending strength is proportional to that value.

I have done that math for you:


a flat 2x4 has about 20% the strength as a standing one
a flat 2x6 has about 7% the strength as a standing one
a flat 2x8 has about 4% the strength as a standing one


You can see that especially for the deeper joists you lose practically all of the strength.