This is challenging - I will just type my own personal feelings on each design here from my experience. I have a ton of experience with them all, and I love them all.
P. works great, works with tons of music styles, more present than the other two if everything is turned to max (more lower mids).... sits well in mix without overpowering.
J. works great, works with tons of music styles, with all knobs cranked has a mid scoop that sits great in a mix but makes bass a little more magically "mixed" sounding, but turn down the bridge pickup volume a bit and it sounds sort of like a precision but with some extra complexity to the tone. sits well in mix without overpowering.
S. works great, works with tons of music styles, by default is a more hifi "pre-mixed" tone with a bit of a scoop in it (or a lot of a scoop in it if it's a 2 band, unless you really mess with the controls and turn up volume on amp). One of the only basses I've owned that I can plugin to board directly and record, no comp, no eq/processing in mix, and it sounds finished. Drawback is that if you have a 2 band it isn't possible to bump the mids without turning down overall level drastically, so once you get your mix tone, you stay with your mix tone, although you can adjust to suit within that range very effectively. It's probably my favorite of the tones for playing rock bass, but I no longer play them full time because I got tired of not being able to get the P style lower mid forward tones, and J style upper mid forward tones, that you can get with nearly every other bass design. Yes, I know, 3 band stingrays exist.... I only have owned 2 bands. Aside from that, just like the fenders, a stingray sits in the mix without overpowering.
I personally feel that a Precision gives the most touch control over your tone (like how you plya the strings, where you pluck, etc). Jazz is less touch controlled but offers the most variation in fundamental tone through knobs.... nothing like 2 pickups to alter your tone. Then the stingray offers by far the most drastic changes to it's tone through it's amazing preamp, one of my 2 favorite preamps in history... but all of those changes are only altering it's one excellent fundamentally slightly scooped tone with that magical treble and artificially enhanced, but amazing, low end.
If I could only own one, and was playing in a band, I'd be happy with any of them, but if I had to go direct only, I'd take the Ray - otherwise either a p or a j would be wonderful.
They are all excellent. J and S can sound quite similar. J and P can sound quite similar. P and S don't really sound all that similar... but do share the "single pickup" tone to a certain degree.