We use scanning tunneling microscopy (STM) and core and valence photoemission as well as low-energy electron diffraction to characterize recently discovered S/Cu(111) surface structures that appear at low coverage below ordering temperatures of around 230 K. At even lower coverage ordered local arrangements are observed near steps and dislocations. Of the laterally extending structures one is open. and honeycomb (hc) like, while three other structures (I,II,III) are more complicated. It is suggested that the structures can be explained as reordered (0001) planes of CuS. Surprisingly the open hc structure gives room for the Cu(111) surface state according to photoemission and scanning tunneling spectra. Core level spectra provide support for one of the models proposed for an earlier studied room-temperature structure [Cu(111)-( root 7x root7)R+/-19.1 degrees -S].