Never use black or white for rigging. These colors create an extreme contrast or visibility that makes them look even bigger than they are. If we are talking about modern ships that have steel rigging, a medium gray makes a much better color. I am excepting old sailing ships, of course, where the standing rigging was tarred, but few of them are in 1:350.
Also, "regular" sewing thread doesn't give an idea of the diameter of the thread. It is getting extremely hard to find sewing thread in various, especially fine, diameters. Craft stores have told me that sewing as a hobby is in decline, and they are not stocking as many threads as they used to.
For 1:700 and small 1:350 (destroyers, patrol boats, etc). I usually use the transparent monofilament thread. It is the smallest diameter I can find locally (it is about ten mil) and the transparent quality makes it look even smaller than it is.
For really fine diameter threads these days I am usually forced to order from places that cater to sailing ship modelers, like Bluejacket and Model Expo. Even they do not have a good, fine medium gray, however.
A not about monofilament thread for rigging. It looks beautiful when done, but is a bear to work with. It is so stiff it is very hard to tie knots in it, so I have to drill terminal areas and glue it.