In this paper, we determine the maximally stable, rotationally invariant domains on the catenoids $\cC_a$ (minimal surfaces invariant by rotations) in the Heisenberg group with a left-invariant metric. We show that these catenoids have Morse index at least $3$ and we bound the index from above in terms of the parameter $a$. We also show that the index of $\cC_a$ tends to infinity with $a$. Finally, we study the rotationally symmetric stable domains on the higher dimensional catenoids.