  The solutions of the one-dimensional homogeneous nonlinear Boltzmann equation
are studied in the QE-limit (Quasi-Elastic; infinitesimal dissipation) by a
combination of analytical and numerical techniques. Their behavior at large
velocities differs qualitatively from that for higher dimensional systems. In
our generic model, a dissipative fluid is maintained in a non-equilibrium
steady state by a stochastic or deterministic driving force. The velocity
distribution for stochastic driving is regular and for infinitesimal
dissipation, has a stretched exponential tail, with an unusual stretching
exponent $b_{QE} = 2b$, twice as large as the standard one for the
corresponding $d$-dimensional system at finite dissipation. For deterministic
driving the behavior is more subtle and displays singularities, such as
multi-peaked velocity distribution functions. We classify the corresponding
velocity distributions according to the nature and scaling behavior of such
singularities.
