Is there any (conjectural) characterization of $\overline{\bf{Q}}$-points on Shimura varieties?

The question of course does not always make sense for ${\bf{Q}}$-points: a theorem of Shimura shows that a quaternionic Shimura curve has no ${\bf{R}}$-points, and a theorem of Mazur shows that the modular curve $Y_0(N)$ has no ${\bf{Q}}$-points for $N$ sufficiently large. The question does however seem to make sense for certain abelian extensions of CM fields. (For instance, in the setting of a quaternionic Shimura curve $M$ defined over a totally real field $F$, if a totally imaginary quadratic extension $K/F$ embeds into the underlying quaternion algebra, then there is a supply of CM points on $M$ defined over certain abelian extensions of $K$). In particular, I should like to know more about the following questions:

(i) Over which number fields $k$ does a given Shimura variety $S(G, X)$ have a $k$-rational point?

(ii) For which such number fields $k$ will $S(G, X)(k)$ be Zariski dense?

(iii) To what extent are such $k$-rational points accounted for by CM points (or similar constructions)?

Sorry if these questions are imprecise or wrongly formulated, I would be happy to at least have an indication of where to look in the literature if someone has already thought about this.