Recently, the prospects for amorphous phases of graphene (α-g) have been explored computationally. Initial models were flat, and contained odd-member rings, while maintaining threefold coordination and sp2 bonding. Upon relaxation, puckering occurs, and may be traced to the existence of pentagons, in analogy with the situation for fullerenes. In this work, we systematically explore the inherent structures with energy close to the flat starting structure. As expected, the planar symmetry can be broken in various ways, which we characterize for 800-atom model of α-g, always using local basis density functional techniques. The classical normal modes of various structural models are discussed, with an emphasis on imaginary modes indicating the evolution from flat to puckered. We also discuss very low energy conformational fluctuations akin to those seen previously in amorphous silicon, and reflect on the nature of the amorphous “ground state” within a network of fixed topology. For completeness, high energy modes were also computed, and are found to be associated with strained parts of the network.