Let $M$ be a finitely generated module over the commutative noetherian ring $R$. Let ${\cal C}(M)$ be the set of all primes $P$ in $R$ such that $R/P$ appears as a quotient in every composition series for $M$. Clearly ${\cal C}(M)$ includes all the primes associated to $M$. But it can contain other primes as well. For example, let $R$ be the local ring of a singular point on an irreducible curve and let $M$ be the maximal ideal of $R$. Then $R/M$ appears as a quotient in every composition series for $M$, but $M$ (as an ideal) is not associated to $M$ (as a module).


So: Is there a nice characterization of the set ${\cal C}(M)$?



Primes that must occur in every composition series for a given module
I think this is a subtle question. The best result I am aware of is the following paper "Filtrations of Modules, the Chow Group, and the Grothendieck Group", by Jean Chan. She proved the following: Let $\mathcal F$ be any composition series of $M$. Let $c\_i(\mathcal F)$ be the formal sum of primes of height $i$. Then $c\_0(\mathcal F), c\_1(\mathcal F)$, as elements in the *Chow group* of $R$, does not depend on the composition $\mathcal F$, so we can talk about $c\_0(M), c\_1(M)$.


So, for example, any such series will always contain the same minimal primes. As for height one primes, we only know that the sum of them is constant up to *rational equivalences*. However, if there is only one height one prime (your example of local ring of curves), then you can deduce its presence in any series by showing $c\_1(M) \neq 0$ in the Chow group of $R$.


I will also note that in Eisenbud's "Commutative Algebra..." after Proposition 3.7, the author remarks that modules which always have a filtration consisting only the associated primes are called *clean*. Clearly for clean modules one has $C(M) = Ass(M)$. There is no clean(!) criterion for cleanliness, as far as I know, but see this [paper](https://arxiv.org/abs/math/0502282) for some partial results!


