I use a very fine thread, a light to medium gray in color. While it is shiny when new, steel cable oxidizes and gets dusty and dirty quickly. It turns to a gray color. It is getting harder to find really thin thread these days, but some specialty sewing shops still carry it.
For 1:700 I use that clear monofilament stuff. Absolutely miserable to work with, but it looks neat.
It is very hard to get rigging to look thin enough. Scale size is almost unobtainably thin. So, reducing contrast is a trick to make it look thinner. Do NOT use black or white. They stand out too much and look thicker than they really are.