I often hear people saying that they are neutral runners, and so they can't tolerate any stability whatsoever in their shoes. Instead, they insist they need a purely neutral shoe with absolutely no stability features. Then they go buy a shoe like the Brooks Ghost 3 and say they bought the most neutral shoe they were able to find. If that shoe's as neutral as you get, I wonder what they would say if they tried on the Saucony Kinvara, Nike Free, or, God forbid, the Vibram FiveFingers (just kidding, VFF fans). Even a shoe like the Nike Pegasus is a small step down in stability. Don't get me wrong, the Ghost is a fine shoe, and it's one of the more neutral trainers in Brooks' line, but it's certainly not the most neutral shoe out there.
Stability, like cushioning, isn't divided into neat little categories, and all shoes classified as "neutral" (or "support" or "motion control") do not have the same amount of stability. On one end of the spectrum, you have ultralightweight flats, like the Mizuno Universe and Asics Piranha, most spikes, like the Nike Zoom Victory and Brooks The Wire, and some of the so-called barefoot shoes, like the Somnio Nada and New Balance Minimus. On the other end, you have the most controlling motion control shoes, like the Brooks Beast and Mizuno Wave Renegade. Everything else lies somewhere in the middle. All of the shoes have some measure of inherent stability, but some have more stability than others. Companies try to draw lines in this spectrum to help the consumer, but it's important to understand that shoe support isn't black and white, but a million shades of grey.
For a while, if a company put a medial post or varus wedge in a shoe, it was automatically bumped out of the neutral category and called either a support or motion control shoe. However, not all medial posts are created equally, and therefore, depending on the size and density of the post (and therefore how much foot motion it controls), shoes within the support and motion control categories span a wide spectrum ranging from mild support to full-on motion control. Additionally, within the neutral category, different shoes offer different amounts of stability based on features like plastic plates in the midfoot (Brooks' is DRB Accel...I don't know what other companies name theirs), heel counter firmness, different flex groove configurations, different midsole materials, and even fabric overlays in the upper. And now, some companies, like Nike with their Lunar line, for example, are offering shoes with dynamic stability, that offer differing levels of stability based on your footstrike (and even within the Lunar line there is still a spectrum of stability, with shoes like the LunarEclipse being more stable than the LunarFly, despite the fact that neither is posted).
While companies often classify their shoes as "neutral," "support," or "motion control," those classifications are rough guidelines in comparison to other shoes in that company's line, intended to help the consumer narrow down their choices. Think of them like ratings on a ski slope. A green circle will be easier than a blue square, which is easier than a black diamond. However, two blues won't be exactly equal in difficulty, and it's almost certain that one will be more difficult than another, though both will be more difficult than a green and easier than a black. And oftentimes, there's not much difference between a hard blue and an easy black. Additionally, even though all resorts use the same grading system, they also grade their runs against other runs on that same mountain. Therefore, it's possible that a blue square at Vail, CO will be comparable to a black diamond at Big Boulder in Pennsylvania. Same deal with running shoes. When you take two neutral shoes, one will almost certainly be more stable than the other, and it's possible that one company's support shoe may actually offer a similar level of support as another company's neutral shoe, especially with the introduction of some of the mild stability shoes like the Saucony Mirage and Nike LunarGlide. Remember, it's a spectrum, and there is some overlap. While companies try to pigeonhole their shoes into neat little categories, they only do that for simplicity's sake, because it doesn't work that way in the real world. For this reason, runners with a neutral gait pattern can usually get away with what some companies classify as mild support shoes, just like mild overpronators are often alright with more stable neutral shoes.