A lot of my thoughts have already been stated here, but I will try to add my 2 cents without being overly repetitive. (BTW, I currently own a Stingray, a J-bass, and a P-bass.)
Stingrays used in jazz: I think they're versatile enough that you can use them for pretty much any style. And, quite frankly, jazz music could stand to occasionally have a more aggressive sounding bass in the mix.
Cost: Obviously, it's up to you to determine if you can afford it, but there are quality basses to be had for a little less money, so don't assume that you have to start at $1,500 to get a decent one. Several brands have already been mentioned.
Visibility: Don't think so much about how many you currently see, but consider how many and who has played them (and the different genres represented) over the past 35+ years. There are quite a few heavyweights on that list.
One final word... If you are inclined at any point to look at a Classic model, inspect it carefully. I had a bad experience with quality on a CS I bought during Guitar Center's recent blowout, and have heard of similar concerns here on TB.
Good luck.