Nylon "expansion" anchor in drywall
Your post contains a picture that shows exactly how this will perform in drywall, and normally it works very well.  

The first 3/8"-1/2" (closest to the screw head) of the anchor does not expand at all, so it won't damage the drywall.  The next 1/2" of the anchor is designed to twist and "knot up" in a hollow cavity, and the tip is designed to slightly expand, but hold the screw securely.

One very important detail that gives these types of anchors a bad name, is that they must be installed into a cleanly drilled hole of the exact right size.  Many people are sloppy with the holes, or just pick a drill bit that looks about the right size.

When the wrong sized hole is used, the "teeth" around the brand name in the picture don't hold strongly in the drywall, and the rest of the anchor will not twist and knot up properly in the wall.  The entire anchor could start twisting which will give the whole installation a loose feeling.

So, follow all the instructions exactly, and drill a clean hole in the right spot the first time, and enjoy.