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][1] for some partial results!

 [1]: https://arxiv.org/abs/math/0502282