  Two-dimensional (2D) Stiefel-Whitney (SW) insulator is a fragile topological
state characterized by the second SW class in the presence of space-time
inversion symmetry. So far, SWIs have been proposed in several electronic
materials but seldom in phononic systems. Here we recognize that a large class
of 2D buckled honeycomb crystals termed Xenes and their ligand-functionalized
derivatives realize the nontrivial phononic SW topology. The phononic SWIs are
identified by a nonzero second SW number $w_2=1$, associated with gaped edge
states and robust topological corner modes. Despite the versatility of
electronic topological properties in these materials, the nontrivial phononic
SW topology is mainly attributed to the double band inversion between in-plane
acoustic and out-of-plane optical bands with opposite parities due to the
structural buckling of the honeycomb lattice. Our findings not only reveal an
overlooked phononic topological property of 2D Xene-related materials, but also
afford abundant readily synthesizable material candidates with simple phononic
spectra for further experimental studies of phononic SW topology physics.
