This is not an answer about the structure that the stability manifold can have in general. However, here is an example (a family of them actually) of such a manifold for an interesting scheme, The Hilbert Scheme of points in $\mathbb{P}^2$. (and stability conditions.) Intuitively, this will say that the stability manifold has the same information as the cone of effective divisors.

Here are more details.

Let $X=\mathbb{P}^{2[n]}$ be the Hilbert scheme of points in $\mathbb{P}^2$. The dimension of $X$ is $2n$ and a theorem by Forgaty claims that $X$ is smooth and $Pic(X)\otimes \mathbb{Q}$ has always rank $2$ (independent of the number of points). Consequently the cone of effective divisors ($Eff(X)$) is always a cone in the plane. It is not hard to prove that such a cone $Eff(X)$ has finitely many chambers(subcones) depending solely on the birrational geometry of $X$.

Keep in mind that cones in the plane ($X_1,X_2$) look like the shaded regions here. (ignore the circle $Y$.)

Now, here comes the birational geometry background. It is true that for each chamber in the cone $Eff(X)$, we can get a birational model $X_D$ (possibly equal to $X$ but in most cases $X_D\ne X$ and $X_D$ birrational to $X$). The seconds paragraph above says that we have finitely many birrational models of $X$. Now here is the question:

**Do these birationational models $X_D$ correspond to ``vary'' the stability condition in the Hilbert scheme of points $X=\mathbb{P}^{2[n]}$?** (meaning, varying a point in the stability manifold.)

Put it more precisely,

**Is there a modular interpretation for the birrational models $X_D$ in terms of moduli spaces of Bridgeland semi-stable objects?**

The answer is yes. Here is a paper by Arcara, Bertram, Coskun and Huizenga where you can find the details.

Roughly speaking then, this paper gives us understanding of the Bridgeland stability manifold (rather a one-dimensional section of it) as a cone in the plane where changing the stability condition (varing a point on the stability manifold) of the Hilbert points corresponds to undergo a birational transformation (and consequently getting a $X_D$.)