  This is the second paper in a series using data from tens of thousands S0
galaxies of the local Universe ($z\lesssim 0.1$) retrieved from the NASA-Sloan
Atlas. It builds on the outcomes of the previous work, which introduced a new
classification scheme for these objects based on the principal component
analysis (PCA) of their optical spectrum and its projections on to the first
two eigenvectors or principal components (the PC1$\unicode{x2013}$PC2 diagram).
We provide a comprehensive characterization of the activity of present-day S0s
throughout both the broad-band PC1$\unicode{x2013}$PC2 spectral classifier and
the conventional narrow-line BPT/WHAN ones, contrasting the different types of
activity classes they define, and present an alternative diagram that exploits
the concordance between WHAN and PCA demarcations. The analysis is extended to
the mid-infrared, radio and X-ray wavelengths by crossmatching our core sample
with data from the WISE, FIRST, XMM$\unicode{x2013}$Newton, and Chandra
surveys. This has allowed us to carry out a thorough comparison of the most
important activity diagnostics in the literature over different wavebands,
discuss their similarities and differences, and explore the connections between
them and with parameters related to star formation and black hole accretion. In
particular, we find evidence that the bulk of nebular emission from radio and
X-ray detected S0$\unicode{x2013}$Seyfert and LINER systems is not driven by
star birth, while the dominant ionising radiation for a number of LINERs might
come from post-AGB stars. These and other outcomes from the present work should
be transferable to other morphologies.
