Atlas 5/8” is my personal favorite for negative rigging. It is miserable for pulling (it’s like a rubber band). All atlas and higher strand sheathed lines glaze badly when used natural crotch, in my experience.
We use 1/2” double esterlon for general use, pulling, and lifting. The 9/16” polydyne comes out for trees tall enough to take it really big.
I like stable braid for a low stretch line (love the 3/8” for mini rigging), but we splice both ends of our rigging lines. In my experience, Samson products tend to milk a lot. Haven’t had a problem with a double-spliced Yale or Sterling line yet.