  We survey fluorescent H2 emission in HST/STIS spectra of the classical T
Tauri stars (CTTSs) TW Hya, DF Tau, RU Lupi, T Tau, and DG Tau, and the
weak-lined T Tauri star (WTTS) V836 Tau. From each of those sources we detect
between 41-209 narrow H2 emission lines, most of which are pumped by strong
Ly-alpha emission. H2 emission is not detected from the WTTS V410 Tau. The
fluorescent H2 emission appears to be common to circumstellar environments
around all CTTSs, but high spectral and spatial resolution STIS observations
reveal diverse phenomenon. Blueshifted H2 emission detected from RU Lupi, T
Tau, and DG Tau is consistent with an origin in an outflow. The H2 emission
from TW Hya, DF Tau, and V836 Tau is centered at the radial velocity of the
star and is consistent with an origin in a warm disk surface. The H2 lines from
RU Lupi, DF Tau, and T Tau also have excess blueshifted H2 emission that
extends to as much as -100 km/s. The strength of this blueshifted component
from DF Tau and T Tau depends on the upper level of the transition. In all
cases, the small aperture and attenuation of H2 emission by stellar winds
restricts the H2 emission to be formed close to the star. The Ly-alpha and the
H2 emission blueshifted by 15 km/s relative to RU Lupi are extended to the SW
by 0.07 arcsec, although the faster H2 gas that extends to about 100 km/s is
not spatially extended. We also find a small reservoir of H2 emission from TW
Hya and DF Tau consistent with an excitation temperature of about 2.5x10^4 K.
