  The James Webb Space Telescope (JWST) will offer the first opportunity to
characterize terrestrial exoplanets with sufficient precision to identify high
mean molecular weight atmospheres, and TRAPPIST-1's seven known transiting
Earth-sized planets are particularly favorable targets. To assist community
preparations for JWST, we use simulations of plausible post-ocean-loss and
habitable environments for the TRAPPIST-1 exoplanets, and test simulations of
all bright object time series spectroscopy modes and all MIRI photometry
filters to determine optimal observing strategies for atmospheric detection and
characterization using both transmission and emission observations. We find
that transmission spectroscopy with NIRSpec Prism is optimal for detecting
terrestrial, CO2 containing atmospheres, potentially in fewer than 10 transits
for all seven TRAPPIST-1 planets, if they lack high altitude aerosols. If the
TRAPPIST-1 planets possess Venus-like H2SO4 aerosols, up to 12 times more
transits may be required to detect atmospheres. We present optimal instruments
and observing modes for the detection of individual molecular species in a
given terrestrial atmosphere and an observational strategy for discriminating
between evolutionary states. We find that water may be prohibitively difficult
to detect in both Venus-like and habitable atmospheres due to its presence
lower in the atmosphere where transmission spectra are less sensitive. Although
the presence of biogenic O2 and O3 will be extremely challenging to detect,
abiotically produced oxygen from past ocean loss may be detectable for all
seven TRAPPIST-1 planets via O2-O2 collisionally-induced absorption at 1.06 and
1.27 microns, or via NIR O3 features for the outer three planets. Our results
constitute a suite of hypotheses on the nature and detectability of
highly-evolved terrestrial exoplanet atmospheres that may be tested with JWST.
