  In this paper we give an upper bound of the first eigenvalue of the Laplace
operator on a complete stable minimal hypersurface $M$ in the hyperbolic space
which has finite $L^2$-norm of the second fundamental form on $M$. We provide
some sufficient conditions for minimal hypersurface of the hyperbolic space to
be stable. We also describe stability of catenoids and helicoids in the
hyperbolic space. In particular, it is shown that there exists a family of
stable higher-dimensional catenoids in the hyperbolic space.
