  We study the omega-limit sets $\omega_X(x)$ in an isolating block $U$ of a
singular-hyperbolic attractor for three-dimensional vector fields $X$. We prove
that for every vector field $Y$ close to $X$ the set $ \{x\in U:\omega_Y(x)$
contains a singularity$\}$ is {\em residual} in $U$. This is used to prove the
persistence of singular-hyperbolic attractors with only one singularity as
chain-transitive Lyapunov stable sets. These results generalize well known
properties of the geometric Lorenz attractor \cite{gw} and the example in
\cite{mpu}.
