This, in view of the analogies between CM points on Shimura curves and torsion points on elliptic curves, is a sequel to an earlier question I had asked: The torsion point count in higher dimension .

Mazur proved that the number of $\mathbb{Q}$-rational torsion points of an elliptic curve is bounded by $16$. Recall in comparison that the modular curve $\mathcal{A}_1 = X_0(1) = \mathbb{A}_j^1$ has exactly $13$ CM points rational over $\mathbb{Q}$; those correspond to the $\bar{\mathbb{Q}}$-isomorphism classes of CM elliptic curves defined over $\mathbb{Q}$. (In addition to the full integer rings of the nine quadratic imaginary fields of class number one, there are also the orders of conductor $2$ in $\mathbb{Q}(\sqrt{-1}), \mathbb{Q}(\sqrt{-3}), \mathbb{Q}(\sqrt{-7})$ and the order of conductor $3$ in $\mathbb{Q}(\sqrt{-3})$.)

It was observed by Will Sawin in the other question that under standard conjectures generalizing Serre's open image theorem, the number of torsion points of degree $\leq d$ over $\mathbb{Q}$ in an abelian variety should be proportional to $d^{\alpha}$ for some exponent $\alpha$ depending on the Mumford-Tate group. For elliptic curves, the two possible exponents are $3/2$ and $2$.

Consider now the same question for the special points $x \in X$ of a Shimura variety having $[\mathbb{Q}(x):\mathbb{Q}] \leq d$. (The special points are the zero-dimensional subvarieties of Hodge type, or the points whose Mumford-Tate group is a torus. ) What should be expected of the asymptotic count of these points as $d \to \infty$? Are they again proportional to a power of $d$? Any good estimates in interesting special cases such as the modular curves $X_0(N)$?