  Monolayer islands of molybdenum disulfide (MoS$_2$) on Au(111) form a
characteristic moir\'e structure, leading to locally different stacking
sequences at the S-Mo-S-Au interface. Using low-temperature scanning tunneling
microscopy (STM) and atomic force microscopy (AFM), we find that the moir\'e
islands exhibit a unique orientation with respect to the Au crystal structure.
This indicates a clear preference of MoS$_2$ growth in a regular stacking
fashion. We further probe the influence of the local atomic structure on the
electronic properties. Differential conductance spectra show pronounced
features of the valence band and conduction band, some of which undergo
significant shifts depending on the local atomic structure. We also determine
the tunneling decay constant as a function of the bias voltage by a
height-modulated spectroscopy method. This allows for an increased sensitivity
of states with non-negligible parallel momentum $k_\parallel$ and the
identification of the origin of the states from different areas in the
Brillouin zone.
