We experimentally demonstrate laser spectroscopy of the electric-quadrupole transition 6s 2S1/2-5d 2D3/2 of a single trapped 171Yb ion. The probed transition has a natural linewidth of 3.1 Hz, and is at a wavelength of 435.5 nm. A 171Yb ion is laser cooled to the range of the Doppler limit in a radio-frequency Paul trap. The F=0-F'=2, mF'=0 hyperfine component of the 2S1/2-2D3/2 transition is excited by a laser source whose frequency is controlled by an optical reference cavity of high intrinsic stability. The recoilless carrier component of the vibrational sideband spectrum is resolved with a minimum linewidth of approximately 80 Hz, corresponding to a fractional frequency resolution of 1.2×10-13. This resolution limit is attributed to laser frequency instability. The effective decoherence rates of the residual ion motion are inferred from the observation that the axial and two radial low-frequency first-order secular-vibration sidebands show minimum linewidths in the range of 0.2-1 kHz. Prospects for an optical frequency standard based on the investigated atomic system are discussed.