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:
$A$ is scattered;
the spectrum of any self-adjoint element in $A$ is countable
all maximal commutative C*-subalgebras of $A$ have a scattered Gelfand spectrum;
the Gelfand spectrum of each commutative C*-subalgebra of $A$ is a Stone space;
every commutative C*-subalgebra of $A$ is generated by its projections;
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 $C$);
Every separable commutative C*-subalgebra of $A$ is AF;
Every C*-subalgebra of $A$ has real rank zero;
Every positive functional on $A$ can be written as a countable sum of pure functionals;
every non-degenerate representation of $A$ is unitarily equivalent with a subrepresentation of a sum of irreducible representations;
The enveloping von Neumann algebra of $A$ is isomorphic to $\prod_{i\in I}B(H_i)$
$A$ 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 $K(H)+1_H\mathbb{C}$.