I use 7/64.
I have a sling on a metal hammock stand. The sling is about 1.5' long. It has a 3" bury. I've used it every night for 3 months. Has not failed once. This sling is short. I made it to replace a foot of 1" link chain because I wanted more options for adjustment. With the chain every adjustment was an inch in or an inch out. With the sling it could be a hair.
I'm 170, 5'4".