  In this paper, we study the stability of catenoids and helicoids in the
hyperbolic $3$-space $\mathbb{H}^3$.
  (1) For a family of spherical minimal catenoids $\{\mathcal{C}_a\}_{a>0}$ in
$\mathbb{H}^3$, there exist two constants $0<a_c<a_l$ such that
  $\bullet$ $\mathcal{C}_a$ is an unstable minimal surface with index one if
$a<a_c$,
  $\bullet$ $\mathcal{C}_a$ is a globally stable minimal surface if
$a\geq{}a_c$, and
  $\bullet$ $\mathcal{C}_a$ is a least area minimal surface in the sense of
Meeks and Yau if $a\geq{}a_l$.
  (2) For a family of minimal helicoids
$\{\mathcal{H}_{\bar{a}}\}_{\bar{a}\geq{}0}$ in $\mathbb{H}^3$, there exists a
constant $\bar{a}_c=\coth(a_c)$ such that
  $\bullet$ $\mathcal{H}_{\bar{a}}$ is a globally stable minimal surface if
$0\leq\bar{a}\leq\bar{a}_c$, and
  $\bullet$ $\mathcal{H}_{\bar{a}}$ is an unstable minimal surface with index
infinity if $\bar{a}>\bar{a}_c$.
