Shoelace type thickness works well.. I also tie an overhand knot as close to the sinker as I can if using nylon so I can just tie on a new sinker and not even a new sinker line for the first few losses. I'm no pro, still figuring out the scratching but that has helped me. What I've found also helped is to make sure my bait is vas and then when stuck, just wait patiently for a decent fish to pull me loose.
Oh, I needed to refresh, you guys beat me to it!
Also there are a few designs of DIY snagless sinkers on the web that I'm sure work better than the standard fare.. As far as I can tell the string helps the sinker ride out of snags easier instead of getting jammed, this has been taken further on the web with sinkers inside paracord but I haven't tried it yet..