  We perform high resolution spectroscopy on $^{176}$Lu$^+$ including the
$^1S_0\leftrightarrow{^3}D_1$ and $^1S_0\leftrightarrow{^3}D_2$ 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.
