ELNES analysis seems to be the most powerful method to identify functionalized graphene single-layers, see for instance:
César Leyva-Porrasa, C. Ornelas-Gutiérrezc,M. Miki-Yoshidac,Yazmín I. Avila-Vegaa, Javier Macossayd, José Bonilla-Cruza: EELS analysis of Nylon 6 nanofibers reinforced with nitroxide-functionalized graphene oxide, Carbon,Vol. 70, April 2014, P. 164–172
TEM is used for structure analysis and Graphene Oxide have disorder structure. The diffraction pattern, crystal size and lattice constant could be helpful at some extent to estimate the quality of Graphene Oxide.
Depending on the capabilities of the instrument a huge amount can be obtained. A recent paper by Krivanek et al. (http://iopscience.iop.org/1742-6596/522/1/012023/pdf/1742-6596_522_1_012023.pdf) demonstrated energy resolution on the order of meV. This would be sufficient to obtain information on the electronic structure. The biggest problem might be beam damage that would alter the signal collected.