  We report a characterization of surfaces of the dichalcogenide TaSe$_2$ using
scanning tunneling microscopy and spectroscopy (STM/S) at 150 mK. When the top
layer has the 2H structure and the layer immediately below the 1T structure, we
find a singular spatial dependence of the tunneling conductance below 1 K,
changing from a zero bias peak on top of Se atoms to a gap in between Se atoms.
The zero bias peak is additionally modulated by the commensurate $3a_0 \times
3a_0$ charge density wave of 2H-TaSe$_2$. Multilayers of 2H-TaSe$_2$ show a
spatially homogeneous superconducting gap with a critical temperature also of 1
K. We discuss possible origins for the peculiar tunneling conductance in single
layers.
