Depends a lot on scale. The biggest problem is that in smaller scales, one cannot find thread of small enough diameter to match the size of rigging on the prototype. You need to find something that looks smaller than it really is. The way to do that is the translucent/transparent monofilament thread, which can be found in pretty thin diameter to start with. You need something like this on 1:350 and 1:700 ships. For larger ships, especially sailing ships, which used actual fiber ropes, thread is fine, and available in a number of diameters. Model Expo carries threads aimed at rigging sailing ships.
For aircraft, 1:72 can benefit from the monofilament thread. 1:48 and 1:32 can use thread. A medium gray simulates stranded stainless wire very well.
The big drawback to monofilament thread is that it is very stiff, and hence hard to tie knots in.
Many models rigged with actual wire have rigging that appears oversized. While very thin wire IS out there, it is hard to find these days. Fewer hams and electronic hobbyists wind their own coils now, so finding #35-#40 wire is getting very hard. I have salvaged radios and old TVs, stripping wire from interstage transformers to get fine wire. You won't find fine wire like that at a hardware store (or even Radio Shack these days).