As a result of the latest turn to the conversation, I actually dragged out several pairs of shoes to do an experiment. I wanted to determine at what heel width does the heel start to contribute actual stability to the human platform. No pun intended. Although my experiment doesn't meet any sort of scientific standard and is purely subjective, it might interest some, as this has been a subject of discussion in the past.
My methodology was to walk slowly, stopping often at key points in the stride to get a feel for how much force it took to tip the heel sideways from the normal vertical position. The key points being those times when both heel and forefoot are in contact with the floor. Most of what I discovered is not surprising, but some of it is. I didn't even bother to test stilettos because I think we can all agree that the heel itself very little lateral resistance to tipping. While I don't find them difficult to walk in, that's not universally true.
I started with heels that are 3/4" wide, of which I own many pairs. It's my favorite heel width besides stilettos, being much more practical for everyday outdoor use than is a stiletto. As I suspected, at 3/4", the heel provides very little lateral stability when standing or walking, very similar to a stiletto. At 1" heel width, that's where one can begin to tell that's he for sure not wearing stilettos. By 1 1/4" in width, the heel is a big contributing factor to shoe stability, in that it takes a lot of ankle force to rock one's heel from side to side. Once you get bigger than 1 1/4", it seems like the law of diminishing marginal return kicks in.
Of course, there is nothing really surprising here, and there are a number of other factors which can affect shoe stability, so it's kind of a silly exercise, but I did it anyhow, just to put some numbers on it. The one kind of surprising thing I found it that when the heel cap becomes significantly worn, as many of mine are, that throws off stability more than you'd think. A 1" broad heel with a worn heel cap is significantly easier to move laterally than one with a brand new one. Keep in mind these tests were done on smooth, hard flooring, and may not directly apply to the real world.