  For any projective curve $X$ let $\bar{M}^d(X)$ be the Simpson moduli space
of pure dimension one rank 1 degree $d$ sheaves that are semistable with
respect to a fixed polarization $H$ on $X$. When $X$ is a reduced curve the
connected component of $\bar{M}^d(X)$ that contains semistable line bundles can
be considered as the compactified Jacobian of $X$. In this paper we give
explicitly the structure of this compactified Simpson Jacobian for the
following projective curves: tree-like curves and all reduced and reducible
curves that can appear as Kodaira singular fibers of an elliptic fibration,
that is, the fibers of types $III$, $IV$ and $I_N$ with $N\geq 2$
