Usually people bother with studying moduli spaces of (coherent) torsion free sheaves that live on a topological space $X$. These spaces, actually stacks, are badly behaved topological spaces. Still, away from the singularities, the structure sheaf or any sheaf of $O_x$-modules is a "nice" sheaf in the sense that it is torsion free from what I am given to understand.
My question is, therefore, can there be any torsion sheaf in such moduli spaces and is there an elementary example to see it? I.e. what kind of element $a \neq \in A$, where $A$ is a local ring on the moduli space would lead to something like $m \cdot a=0$ for a $m \neq 0 \in M$, where $M$ is a module (in other words this means that $m$ is a section of the sheaf of $O_M$-modules)?
***I have the impression that no sheaf can be torsion sheaf if the underlying topological space $X$ is smooth.
I think there is some misunderstanding going on here.
You say "I have the impression that no sheaf can be torsion sheaf if the underlying topological space $X$ is smooth", but this is not correct.
On any variety (or scheme) of positive dimension, there are lots of torsion sheaves. In particular, on an integral scheme, the torsion sheaves are exactly the sheaves that are supported on proper closed subsets.