The six-vertex model is an important toy-model in statistical mechanics for two-dimensional ice with a natural parameter D. When D=0, the so-called free-fermion point, the model is in natural correspondence with domino tilings of the Aztec diamond. Although this model is integrable for all D, there has been very little progress in understanding its statistics in the scaling limit for other values. In this talk, we focus on the six-vertex model with domain wall boundary conditions at D = 1/2, where it corresponds to alternating sign matrices (ASMs). We consider the level lines in a height function representation of ASMs. We report that the maximum of the topmost level line for a uniformly random ASMs has the GOE Tracy-Widom distribution after appropriate rescaling and will discuss many open problems related to this model. Much of this talk is based on joint work with Arvind Ayyer and Kurt Johansson.