Indeed, these C*-algebras are scattered as pointed out by Eric Wofsey.
The following might be useful. For a C*-algebra $A$, the following statements are equivalent:

1) $A$ is scattered;

2) the spectrum of any self-adjoint element in $A$ is countable

3) all maximal commutative C*-subalgebras of $A$ have a scattered Gelfand spectrum;

4) the Gelfand spectrum of each commutative C*-subalgebra of $A$ is a Stone space;

5) every commutative C*-subalgebra of $A$ is generated by its projections; 

6) every C*-subalgebra $C\subseteq A$ is AF (but possibly non-separable, i.e. there is a directed set of finite-dimensional C*-subalgebras of $C$ whose union is dense in $C$);

7) Every separable commutative C*-subalgebra of $A$ is AF;

8) Every C*-subalgebra of $A$ has real rank zero;

9) Every positive functional on $A$ can be written as a countable sum of pure functionals;

10) every non-degenerate representation of $A$ is unitarily equivalent with a subrepresentation of a sum of irreducible representations;

11) The enveloping von Neumann algebra of $A$ is isomorphic to $\prod_{i\in I}B(H_i)$ 

12) $A$ does not have a commutative C*-subalgebras whose Gelfand spectrum is homeomorphic to $[0,1]$;

Scattered C*-algebras were defined in the articles "Scattered Algebras" and "Scattered Algebras II" by Jensen, where the equivalence of 9,10 and 11 have been proven. Kusuda proved in "C*-algebras in which every C*-subalgebra is AF the equivalence of all statements involving (commutative) C*-subalgebras. Lin proved in "The structure of Quasi-Multipliers of C*-algebras" that scattered C*-algebras are AF.

An example of a non-commutative scattered C*-algebra is $K(H)$, the compact operators on some Hilbert space $H$. An example of a unital non-commutative scattered C*-algebra is $K(H)+1_H\mathbb{C}$.