What are the possible topological behaviors of an isometrically immersed hyperbolic plane (which we call a geodesic plane) in a hyperbolic 3-manifold? When the 3-manifold has finite volume, every geodesic plane is closed or dense, due to Shah and Ratner independently. This remarkable rigidity can be generalized to certain 3-manifolds of infinite volume: when the 3-manifold is geometrically finite and acylindrical, every geodesic plane intersecting the interior of the convex core is closed or dense there, due to recent works of McMullen-Mohammadi-Oh and Benoist-Oh. In this talk, we will address a few questions arising from their works. First, is a geodesic plane closed in the interior of the convex core closed in the whole manifold? Second, how does a geodesic plane outside the convex core behave? We will give a negative answer to the former with an example. For the latter, we will describe the connection between the problem at hand and the existence of an infinite ray with finite transverse measure with respect to a given measured geodesic lamination, and give a brief introduction of joint work in progress with Tina Torkaman in this regard.