  We study CM points on the Shimura curves $X_0^D(N)_{/\mathbb{Q}}$ and
$X_1^D(N)_{/\mathbb{Q}}$, parametrizing abelian surfaces with quaternionic
multiplication and extra level structure. A description of the locus of points
with CM by a specified order is obtained for general level, via an
isogeny-volcano approach in analogy to work of Clark and Clark--Saia in the
$D=1$ case of modular curves. This allows for a count of all points with CM by
a specified order on such a curve, and a determination of all primitive residue
fields and primitive degrees of such points on $X_0^D(N)_{/\mathbb{Q}}$. We
leverage computations of least degrees towards the existence of sporadic CM
points on $X_0^D(N)_{/\mathbb{Q}}$.
