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 for some partial results!
