  Given positive integers $d\geq\kappa$, and a subset $\Gamma\subset [0,1]$,
let $\mathrm{Ivol}_{\mathrm{lc}}^{\Gamma}(d,\kappa)$ denote the set of Iitaka
volumes of $d$-dimensional projective log canonical pairs $(X, B)$ such that
the Iitaka--Kodaira dimension $\kappa(K_X+B)=\kappa$ and the coefficients of
$B$ come from $\Gamma$. In this paper, we show that, if $\Gamma$ satisfies the
descending chain condition, then so does
$\mathrm{Ivol}_\mathrm{lc}^\Gamma(d,\kappa)$ for $d\leq 3$. In case $d\leq 3$
and $\kappa=1$, $\Gamma$ and $\mathrm{Ivol}_\mathrm{lc}^\Gamma(d,\kappa)$ are
shown to share more topological properties, such as closedness in $\mathbb{R}$
and local finiteness of accumulation complexity. In higher dimensions, we show
that the set of Iitaka volumes for $d$-dimensional klt pairs with Iitaka
dimension $\geq d-2$ satisfies the DCC, partially confirming a conjecture of
Zhan Li.
  We give a more detailed description of the sets of Iitaka volumes for the
following classes of projective log canonical surfaces: (1) smooth properly
elliptic surfaces, (2) projective log canonical surfaces with coefficients from
$\{0\}$ or $\{0,1\}$. In particular, the minima as well as the minimal
accumulation points are found in these cases.
