  We consider the problem of finding embeddings of arc-like continua in the
plane for which each point in a given subset is accessible. We establish that,
under certain conditions on an inverse system of arcs, there exists a plane
embedding of the inverse limit for which each point of a given countable set is
accessible. As an application, we show that for any Knaster continuum $K$, and
any countable collection $\mathcal{C}$ of composants of $K$, there exists a
plane embedding of $K$ in which every point in the union of the composants in
$\mathcal{C}$ is accessible. We also exhibit new embeddings of the Knaster
buckethandle continuum $K$ in the plane which are attractors of plane
homeomorphisms, and for which the restriction of the plane homeomorphism to the
attractor is conjugate to a power of the standard shift map on $K$.
