Let $C$ be an irreducible projective curve with at worst nodal singularities. Let $E$ be the trivial locally free sheaf of rank $r$ i.e., $E$ is the direct sum of $r$ copies of the trivial line bundle on $C$. The question: is $E$ semi-stable?
I would think the answer to the question is yes, since the degree of $E$ is zero which implies the degree of any coherent subsheaf of $E$ is also $0$. In turn the slope of both $E$ and any subsheaf of $E$ is zero, implying semi-stability. Am I right?
A direct sum of line bundles is slope semi-stable if and only if each summand is slope semi-stable and all the slopes agree.
So it's enough to show that the trivial line bundle is slope semi-stable, but all line bundles are slope stable.