For black tie, I usually wear a white linen handkerchief, but I sometimes wear a silk handkerchief of an "interesting" pattern. For white tie I only wear white linen, although the doublet I tend to wear for these events does not have a breast pocket.

My day wear handkerchiefs run the gamut from bandannnas, to silks, to linens.

I usually just "stuff" my handkerchiefs, but sometimes I do fold them. It just depends.