Let $\mathcal{S}$ be a complete hyperbolic surface and consider the horocycle flow on its unit tangent bundle $T^1\mathcal{S}$. Whenever $\mathcal{S}$ is geometrically finite, it is well known that all horocycle orbits are either closed or dense in $\mathcal{E} \subseteq T^1\mathcal{S}$, the non-wandering set for the horocycle flow. We consider $\mathbb{Z}$-covers of compact surfaces, arguably the simplest examples of geometrically infinite surfaces, and study the possible horocycle orbit closures that they carry. We show that the structure of the orbit closures in such covers is delicately dependant on the particular geometry of the covered compact surface and allow for wide variability. We present a construction of a family of surfaces for which a complete explicit orbit closure classification is given. Based on joint work in progress with James Farre and Yair Minsky.