There are quite a few ways to approach this problem. Good hobby shops nowadays carry some really fine wire - fine enough to be convincing ship rigging. Check the railroad department for the brass wire sold by Detail Associates. Another possible source of wire is the multi-strand electrical stuff, which can be separated into individual strands.
That old favorite of plastic modelers, stretched sprue, has its advantages. With a little practice it can be produced in any diameter you like. (Mixing sizes of rigging line is an excellent, easy way to make a model look more authentic. The stays that hold up a mast, for example, should be considerably heavier than a singnal halyard.) Once the stretched sprue is on the model it can be tightened (if you have the nerve) by holding the model upside down over a lighted candle. Start with the model and the candle abou two feet apart, and very slowly bring the model closer. As the heat from the flame reaches the rigging line it will suddenly tighten. A few seconds later it will melt. The trick is to yank the model out of the way at the crucial moment.
Stretched sprue has several other key attributes: it's easy to cut where you want it to, and easy to break where you don't. The latter is a mixed blessing. I don't like to think about how many lines on ship models I've had to replace because somebody has touched them with a little too much enthusiasm, or something has bumped into them. On the other hand, if they'd been wire they probably would have torn down some crucial part of the model when they went.
A few weeks ago I was in the midst of taking some closeup pictures of a 1/700 model and (there are a couple of valuable lessons to be learned here) I turned the tripod around so that one of its legs was directly behind the camera lens. The weight of the lens made the camera fall over on its face. It landed on one of the ship's boat booms, a piece of brass wire with a photo-etched ladder hanging from it and several stretched sprue guy lines connecting it to the hull. The wire and the ladder bent; the sprue broke in several places. The repairs took about fifteen minutes. If that sprue had been wire, the hull might well have been damaged.
My personal experience is that working with stretched sprue has a learning curve that I have to re-experience, to some extent, each time I try it. It seems like the first attempt I make on each model is an exercise in frustration; the stuff comes out the wrong length, or I can't get both ends to stay put simultaneously, or it sags too much, or something. If I force myself to keep at it for an hour or so, though, my fingers seem to get in a groove and things go pretty well.
I've found that a pair of tweezers sold for handling stamp collections is an excellent tool for this sort of rigging work. Such tweezers have ends that are shaped like miniature spatulas, with slightly rounded edges. It's almost impossible to break a piece of sprue inadvertently with them. (For most modeling work I use a pair of sharp-pointed tweezers that I've had for about thirty years. They're among my favorite tools, but they're lousy for sprue work.)
My favorite adhesive for sprue work is Elmer's glue. It shrinks a little as it dries, can be removed with water when it's wet, and, when applied to painted plastic, sticks just well enough. If a piece of sprue has to be replaced later due to damage, it's usually pretty easy to break the glue loose and start over.
These are all just one person's personal experiences; what works for one modeler isn't necessarily the best idea for another. Hope this is of some help, though.