Let $X$ be a smooth complex projective threefold, and $\beta\in H_2(X,\mathbb Z)$ a curve class. In the Kontsevich's moduli space of stable maps, $\overline M_g(X;\beta)$, a general point $[f:C\to X]$ is represented by a map which is an immersion. In some sense this is the closest we can get, in this moduli space, to what a "curve in $X$" should look like.
I was wondering what the situation is in Donaldson-Thomas theory. Here the moduli space of curves $M=M(\textrm{ch})$ is a moduli space of torsion-free sheaves on $X$, with Chern character $\textrm{ch}=(1,0,-\beta,n)$, where $n$ keeps track of both the genus and the number of isolated and embedded points. Curves here are all embedded, but they have some isolated and embedded points in general (their number is $r=p_a-1+n$).
Question. What are the good properties that a general point of $M$ does have?
(Just some random "good" qualities: connected, reduced, Cohen-Macaulay, pure...).
Thank you in advance!