My favorite material for sling cords is hemp.I like it's feel mostly,it's look,it has no stretch but crappy stone wear it quite quickly and sometimes the release knot gets blasted off.Soo as much as i love hemp,i use 3mm synthetic rope similar to paracord quite often to spare my hemp slings. Also braided natural slings have much more control imo because the cords don't twist like a single strand of paracord for ex.  
I also believe in something most would say it doesn't exist or it's jlasud and his crap again : natural materials respond,communicate  much better with humans than materials altered by men.This relation can only be felt,when you use a sling or loose a wooden arrow after an aluminium one,or steps into a wooden house after a plastic one etc.
I also used the advantage of 2mm synthetic cord for making the longest and most aerodynamic sling that couldn't really be made from natural materials(maybe sinew or horsehair).It is 1.5 meters long (59") and with that i can sling child fist size stones ~170m. I feels like artillery

Welcome, Mike;
  No offense taken.  Though I prefer natural materials such as jute, hemp, and cotton many of the slings I make are from synthetics.  I've made slings with synthetic yarn, nylon, paracord, and polypropylene.  I've tried plastic cord but that didn't come out so well.  Also I have an Aussie pouch sling.  My favorite synthetic material is polypropylene.  The stuff I buy is sheathed like paracord.  3/8" polypropylene is perfect.  It's soft and easy to work with and makes a good looking sling that's just about indestructible.  
    As far as slings made from synthetic materials having an advantage over those made with natural materials, other than durability I haven't noticed any.  
   Though I like a rustic looking jute sling as much as anything, there's nothing wrong at all with using synthetic materials IMO.  

One thing I like about slings is that it's really hard to over complicate them. Modern cordage offers ease, and might (?) give you an edge for distance records but otherwise it's really about what appeals to you most as a slinger.

I like to think as a "new" sling as a Glock 19, and an "old" sling like an Ed Brown 1911. The glock is cheaper, easier to come by, and generaly better for new shooters; however an expiereinced shooter can see the tremendous beauty and functionality of a time tested design, though it might not look as "high tech".
 
I use both so I don't want to lean towards one side too much, however I generaly prefer natual material slings. Mainly for aesthetic reasons, but it also shows how even through thousands of years the old design and materials can shoot just as well as the "young" ones.
 
Also, if you stick to slinging for awhile, no matter what your prefference is, you will find yourself with a large collection of slings made from almost every material imaginable.  

Dyneema is one of the new braided fishing lines, super strong for the diameter, 50 pound test dyneema is slightly smaller than 10 pound test monofiliment.  Think Spyderwire.  
 
You can braid a very skinny sling that is very strong, I don't think it handles abrasion very well, so you will need a leather pouch and you will definately need to pad the finger loop and you will probably need a button on the release cord.

Everyone,
 
Thank you for the discussion.  I agree that the Glock is the epitome of the modern firearm and certainly has its place.  Yet sometimes, a .50 cal flintlock is just what is called for.
 
I can see that in fairly short order I'll have a stable full of slings of different styles and materials.
 
The Dyneema I have is 1/8" and is commonly referred to as Amsteel Blue http://www.samsonrope.com/index.cfm?rope=192
 
It has very high strength and abrasion resistance.  It is often used to replace wire rope on sailboats for their standing rigging.
 
I've been using it to hold me up in my hammock so I know it is plenty strong for a sling.  In the next week or so I should be able to get my hands on some good leather.  When I do I'll put some of my Amsteel to use and see how it performs.
 
While I've got your attention, what is a good length for a sling for a figure-8 throwing style?  My current sling is about 33" when I'm holding both ends.
 
Thanks,  
 
Mike

lol actually it's kind of not representative - the braiders are generally more enthusiastic and voluble. Always looking for different patterns and techniques. Whereas us folk who just make paracord and leather slings don't have alot ot say as the techniques are pretty basic.  
 
For example I only use simple overhand knots in my slings - nothing else is needed. So not a lot to talk about  
 
So while in reality there are an awful lot of slings made with modern materials, it's the historic side that generates the most discussion