I've tested a couple guns with Stability. It reduces the amount the crosshairs creep up the screen while firing. There is a maximum creep point too where it will stop. These two elements seems to be the only consequences of "recoil". Spread is not affected. Aiming also reduces "recoil" and can actually reduce it to zero. So if you always aim, Stability is probably a trash mod.
Stability on the P-15 contributes nothing while aiming, when not aiming it reduces the creep by about half (both the rate of creep and the maximum creepage).
However, on the Oppressor it helps even when aiming (though it's not real significant).
As some people have mentioned, "Hip-fire" apparently reduces the amount of random "spread" you get. At some stage, as with "recoil", the "spread" will get to approximately zero and any additional efforts to reduce it further will be wasted, so combining Scope with either of these mods on certain weapons probably isn't the best investment. I think that's the moral of the story here. Also, that "hip fire accuracy" is a bad name. It should be simply "accuracy", it still applies when using a scope, it's just that on some/most/all? guns that's going to make no difference so you better do some science first.
Here's my Stability science, I aimed at the bottom of a sign. Top videos are without the mod, bottom are with it...
P-15 video: Proves to me that Stability is practically useless on the P-15, since you wouldn't normally unload a full clip on a target like this
Oppressor: It's useless unless you don't aim (in which case your Oppressor is useless anyway)