  Measuring the quasi-normal mode~(QNM) spectrum emitted by a perturbed
black-hole~(BH) --~also known as BH spectroscopy~-- provides an excellent
opportunity to test the predictions of general relativity in the strong-gravity
regime. We investigate the prospects and precision of BH spectroscopy in
massive binary black hole ringdowns, one of the primary science objectives of
the future Laser Interferometric Space Antenna~(LISA) mission. We simulate
various massive binary BH population models, featuring competing prescriptions
for the Delays between galaxy and BH mergers, for the impact of supernova
feedback on massive BH growth, and for the initial population of high redshift
BH seeds (light versus heavy seeds). For each of these scenarios, we compute
the average number of expected events for precision BH spectroscopy using a
Fisher-matrix analysis. We find that, for any heavy seed scenario, LISA will
measure the dominant mode frequency within ${\cal O}(0.1) \%$ relative
uncertainty and will estimate at least 3 independent QNM parameters within $1
\%$ error. The most optimistic heavy seed scenarios produce $\mathcal{O}(100)$
events with $1 \%$ measurability for 3 or more QNM quantities during LISA's
operational time. On the other hand, light seed scenarios produce lighter
merger remnants, which ring at frequencies higher than LISA's sensitivity.
Interestingly, the light seed models give rise to a fraction of mergers in the
band of Einstein Telescope, allowing for the measurement of 3 QNM parameters
with $\sim 10 \%$ relative errors in approximately a few to ten events/yr. More
precise BH spectroscopy in the light seed scenarios would require instruments
operating in the deciHertz band.
