  Single ionization of H$_2$ molecules exposed to strong and short laser pulses
is investigated by a semi-classical method. Three laser characteristics are
considered: i) The carrier-wave frequency corresponds to wavelengths covering
and bridging the two ionization regimes: From tunnel ionization (TI) at 800 nm
to multiphoton ionization (MPI) at 266 nm. ii) Values of the peak intensity are
chosen within a window to eliminate competing double ionization processes. iii)
Particular attention is paid to the polarization of the laser field, which can
be linearly or circularly polarized. The results and their interpretation
concern two observables, namely the end-of-pulse total ionization probability
and vibrational distribution generated in the cation H$_2^+$. The most
prominent findings are an increased ionization efficiency in circular
polarization and a vibrational distribution of the cation that favors
lower-lying levels than those that would be populated in a vertical
(Franck-Condon) ionization, leading to non Franck-Condon distributions, both in
linear and circular polarizations.
