  The inelastic scattering of electrons which carry current through a
single-molecule junction is modeled by a quantum dot, coupled to electron
reservoirs via two leads. When the electron is on the dot, it is coupled to a
single harmonic oscillator of frequency $\omega_0$. At zero temperature, the
resonance peak in the linear-response conductance always narrows down due to
the coupling with the vibrational mode. However, this narrowing down is given
by the Franck-Condon factor only for narrow resonances. Contrary to some claims
in the literature, the linear-response conductance does not exhibit any
side-bands at zero temperature. Small side-bands, of order $\exp [-\beta \hbar
\omega_0]$, do arise at finite temperatures. The single-particle density of
states exhibits discontinuities and logarithmic singularities at the
frequencies corresponding to the opening of the inelastic channels, due to the
imaginary and real parts of the self-energy. The same singularities also
generate discontinuities and logarithmic divergences in the differential
conductance at and around the inelastic thresholds. These discontinuities
usually involve upwards steps, but these steps become negative within a rather
narrow range of the elastic transparency of the junction. This range shrinks
further as the the excitation energy exceeds the bare resonance width.
