I always carry them in my hold luggage, and I usually have a lot more than one.

As I always travel with at least one knife and a multitool as well as sling making kit and whatever other slinging related items I wouldn't want going through an x-ray machine, I always have hold luggage.

I have had slings in my hand luggage occasionally, generally braided ones with no metal parts - my EDC slings are riveted so don't get risked through x-ray machines.

So, yes slings are not a problem in carry on luggage.

As far as ammo goes, well that depends on which gods watch over you.

Uwe had 20 odd clay glandes he had to pay full hold luggage extra for.
While Roman and guelu travelled halfway across the world with about 20kgs each of guamian beach rocks in their carry on luggage, along with innumerable slings and sling making kit.

But you certainly never need to go anywhere without a sling

I'm like C-A -- I always have a knife and a multi-tool so they always go in hold baggage.  Ditto for slings and usually a few re-useable ammo golf balls.  Easy peasy for international travel because you always have a lot of stuff.  Around the US I almost always fly Allegiant, and since I'm a Veteran I get free hold baggage, which is sweet.

IMHO a sling or slings should not raise any eyebrows with airport security -- as long as you don't go around the airport bragging about how far and hard you can chuck rocks with "this little bitty length of string"... 

If they ask what it is, tell them it's part of a game -- don't mention the W word (I said don't mention the word "weapon").  It's not a weapon, it's a lead for the dog you're flying to take care of; or part of a beach game, a piece of mountaineering gear you forgot to pack; or a headband for your hippy girlfriend; or a sex toy.... anything but a rock throwing sling.

I had it in my pocket. It was Christmas time I was going to say it is something to hang on my tree but no one ever asked.