We study the existence of plane substructures in drawings of the $d$-dimensional hypercube graph $Q_d$. We construct drawings of $Q_d$ which contain no plane subgraph with more than $2d-2$ edges, no plane path with more than $2d-3$ edges, and no plane matching of size more than $2d-4$. On the other hand, we prove that every rectilinear drawing of $Q_d$ with vertices in convex position contains a plane path of length $d$ (if $d$ is odd) or $d-1$ (if $d$ is even). We also prove that if a graph $G$ is a plane subgraph of every drawing of $Q_d$ for a sufficiently large $d$, then $G$ is necessarily a forest of caterpillars. Lastly, we give a short proof of a generalization of a result by Alpert et al. [Cong. Numerantium, 2009] on the maximum rectilinear crossing number of $Q_d$.