There is a risk associated with cotton rope because many birds will chew and shred it, making long fibres that can get wound tightly around legs and toes. There's also the risk of ingestion and the formation of an intestinal obstruction from a ball of cotton fibres. Having said this, though, my birds basically ignore cotton rope and so I use it. The risk is there, though.
Sisal is generally thought to be safe because the fibres are short and easily broken. I use all sizes from sisal string for making toys right up to the 1" thick rope for stringing across my cages as a nice swingy perch. If you check on YouTube, it's easy enough to learn how to make an eye-splice (ie. put an indestructible loop on the end of your rope/string) - this opens up many possibilities.
There's another kind of rope called paulie/polly/poly rope which is often sold at petshops and bird-toy suppliers. It's a synthetic rope which doesn't have fibres and is supposedly safe to use. I don't like it because it's too stiff and abrasive to work with and my Beaks can easily chew through it in one go. Other kinds of nylon rope are probably not going to be safe because of the ingestion hazard. In fact, anything with fibres or plies that can be dislodged by an inquisitive bird is probably a bit suss.