We perform high resolution spectroscopy on ^176Lu^+ including the ^1S_0^3D_1 and ^1S_0^3D_2 clock transitions. Hyperfine structures and optical frequencies relative to the ^1S_0 ground state of four low lying excited states are given to a few tens of kHz resolution. This covers the most relevant transitions involved in clock operation with this isotope. Additionally, measurements of the ^3D_2 hyperfine structure may provide access to higher order nuclear moments, specifically the magnetic octupole and electric hexadecapole moments.