The dynamics of horocycle flow in a hyperbolic surface goes back to Hedlund in the 1930's who showed for example that in a closed surface every horocycle is dense. In the infinite-area case, a variety of phenomena can happen. For a Z^d cover of a compact surface S, there is an interesting connection between the behavior of orbit closures and the geometry of Thurston-type stretch laminations in S, as governed by the "stable norm" on the deck group. I will try to tell this story, which I hope is of interest to hyperbolic geometers as well as homogeneous dynamicists. This is joint with James Farre and Or Landesberg, and while some is work in progress, in many cases we have a complete classification of orbit closures and their Hausdorff dimensions.