  We consider a buckled quantum spin Hall insulator (QSHI), such as silicene,
proximity-coupled to a conventional spin-singlet, s-wave superconductor. Even
limiting the discussion to the disorder-robust s-wave pairing symmetry, we find
both odd-frequency ($\omega$), spin-singlet and spin-triplet pair amplitudes
and where both preserve time-reversal symmetry. Our results show that there are
two unrelated mechanisms generating these different odd-$\omega$ pair
amplitudes. The spin-singlet state is due to the strong inter-orbital processes
present in the QSHI. It is exists generically at the edges of the QSHI, but
also in the bulk in heavily doped regime if an electric field is applied. The
spin-triplet state requires a finite gradient in the proximity-induced
superconducting order along the edge, which we find is automatically generated
at the atomic scale for armchair edges but not at zigzag edges. In combination
these results make superconducting QSHIs a very exciting venue for
investigating not only the existence of odd-$\omega$ superconductivity, but
also the interplay between different odd-$\omega$ states.
