We investigate whether it is typical for a sparse graph to be uniquely characterized by its adjacency spectrum up to isomorphism. Our first result shows that the giant component of an Erdős-Rényi graph is cospectral when the average degree is sufficiently small. The proof relies on the existence of a specific pendant tree, combined with a method by Schwenk that swaps trees to construct a cospectral mate. It seems possible that pendant trees are essentially the only obstruction, meaning that the giant should become characterized by spectrum with high probability if one prunes these by considering the 2-core. The majority of the paper is devoted to theoretical and numerical evidence supporting this concept. Our main theorem in this direction establishes that local switching methods can not cause the 2-core to be cospectral. We also discuss R-cospectrality and rational cospectrality at fixed level.