I would warn against a neoprene strap. There's another famous brand makes mostly vinyl straps. Both are well-documented to eat into nitrocellulose not always, but in certain cases. It's the chemical in those materials that keeps them flexy and supple that does it, as I understand it. Can be a bit stinky too.
Another big brand that makes "100% genuine" leather straps, very decorative, look super tasty at attractive price points. Beware of bonded leather, like those offshore belts that flake off top layers or separate inside. It's as much leather as MDF is wood. A lot of chems in that stuff. If you've ever been tricked by bonded leather on a belt, sofa, whatever, you'll understand what I'm saying.
I use vinyl myself sparingly, as a less expensive option sometimes, or more often as a decorative accent material. Boy, it stitches so nice. I use a lot of heavier fabrics too. In fact, sometimes I favour a good heavy fabric to leather, or in combination with leather.
Above all, one should never store any strap in a case against an instrument or hanging against. I used to before I knew better, luckily
no damage. I hardly case my guitars as they hang mostly and seldom leave my home.
Nott at all surprised to see so many here using the bare minimum poly 2" wide starter pack shoulder-hammer straps. Very common. I have two of those myself, here my inventory, that rarely get used anymore. I've made it my mission to preach the "better strap" gospel.
Unless you have seven pounder or wear it on a vest or suit jacket, it's gonna hurt after a while. It's purely a pressure point / surface area thing. When I watch a live band with those in use, I measure how long it takes before the users starts grimacing and shifting in search of relief. Usually about five songs in. Test for it yourself. Performers don't even realize they're doing it. Normally doesn't happen until way later if those straps are worn on top of a suit jacket, a vest or layered clothing, but not long on a button shirt or tee. Also, It's not often you see that same sort of strap on guitars for a whole event. There will usually be a switch-out.
At least a natural material like cotton canvas or hemp can offer a tiny bit of flex. The ubiquitous poly web sealt belt stuff is so unforgiving. Used to be ten, fifteen years ago the eight dollar poly strap was an OK deal. But the few stores kind enough stock some of my goods beside the budget straps, I see those getting cheaper and cheaper by the year, thinner body, sketchier ends. But, boy they sure move. And the margin stores make on those is unbelievable. And I see almost the same thing by a European brand with noteably better real
leather ends, rather than poly web sandwiched by two layers of thin vinyl (it's common) at ten to fifteen times the price. Makin' bank!
Guitarists are fickle creatures. I can say many crave all kinds of metal bits on straps, the more the better! Others? Not going there. Some enjoy big brand names up front or all over the strap, others, NO DAMN WAY! Some players demand a real slippy strap.
Others, it drives 'em nuts. There's those who want all sorts of lacing, grommets, fancy, retro or getting all matchy/matchy with the guitar. others want low-key plain jane. Some want super flexy, some stiff. Wear the skin from a real dead animal? Some will just refuse. Makes it fun and interesting as a builder who also offers custom work.
I think we all need at least one plainish black leather strap, I say 2.5 - 3" wide and a least lightly padded. You can get by most cases,
with just that one. I have to force myself to make those because: super boring. Consider a reptile emboss, or some texture on it.
Also, I'll probably get heat on this, but your primary reason for a dedication to a certain style of strap should not be "It fits nice inside the accessory compartment of my guitar case." Convenient, but you're paying in other ways.