  Classification of possible quantum spin liquid (QSL) states of interacting
spin-1/2's in two dimensions has been a fascinating topic of condensed matter
for decades, resulting in enormous progress in our understanding of
low-dimensional quantum matter. By contrast, relatively little work exists on
the identification, let alone classification, of QSL phases for spin-1 systems
in dimensions higher than one. Employing the powerful ideas of tensor network
theory and its classification, we develop general methods for writing QSL wave
functions of spin-1 respecting all the lattice symmetries, spin rotation, and
time reversal with trivial gauge structure on the square lattice. We find $2^5$
distinct classes characterized by five binary quantum numbers. Several explicit
constructions of such wave functions are given for bond dimensions $D$ ranging
from two to four, along with thorough numerical analyses to identify their
physical characters. Both gapless and gapped states are found. The topological
entanglement entropy of the gapped states are close to zero, indicative of
topologically trivial states. In $D=4$, several different tensors can be
linearly combined to produce a family of states within the same symmetry class.
A rich "phase diagram" can be worked out among the phases of these tensors, as
well as the phase transitions among them. Among the states we identified in
this putative phase diagram is the plaquette-ordered phase, gapped resonating
valence bond phase, and a critical phase. A continuous transition separates the
plaquette-ordered phase from the resonating valence bond phase.
