  Trend filtering---first introduced into the astronomical literature in Paper
I of this series---is a state-of-the-art statistical tool for denoising
one-dimensional signals that possess varying degrees of smoothness. In this
work, we demonstrate the broad utility of trend filtering to observational
astronomy by discussing how it can contribute to a variety of spectroscopic and
time-domain studies. The observations we discuss are (1) the Lyman-$\alpha$
forest of quasar spectra; (2) more general spectroscopy of quasars, galaxies,
and stars; (3) stellar light curves with planetary transits; (4) eclipsing
binary light curves; and (5) supernova light curves. We study the
Lyman-$\alpha$ forest in the greatest detail---using trend filtering to map the
large-scale structure of the intergalactic medium along quasar-observer lines
of sight. The remaining studies share broad themes of: (1) estimating
observable parameters of light curves and spectra; and (2) constructing
observational spectral/light-curve templates. We also briefly discuss the
utility of trend filtering as a tool for one-dimensional data reduction and
compression.
