  The monolayer FeSe with a thickness of one unit cell grown on a
single-crystal SrTiO$_{3}$ substrate (FeSe/STO) exhibits striking
high-temperature superconductivity with transition temperature $T_{c}$ over 65K
reported by recent experimental measurements. In this work, through analyzing
the distinctive electronic structure, and providing systematic classification
of the pairing symmetry , we find that both $s$-and $p$-wave pairing with odd
parity give rise to topological superconducting states in monolayer FeSe, and
the exotic properties of $s$-wave topological superconducting states have close
relations with the unique non-symmorphic lattice structure which induces the
orbital-momentum locking. Our results indicate that the monolayer FeSe could be
in the topological nontrivial $s$-wave superconducting states if the relevant
effective pairing interactions are dominant in comparison with other
candidates.
