  We give a complete analysis of low-velocity dynamics close to grazing for a
generic one degree of freedom impact oscillator. This includes nondegenerate
(quadratic) grazing and minimally degenerate (cubic) grazing, corresponding
respectively to nondegenerate and degenerate {\em chatter}. We also describe
the dynamics associated with generic one-parameter bifurcation at a more
degenerate (quartic) graze, showing in particular how this gives rise to the
often-observed highly convoluted structure in the stable manifolds of
chattering orbits. The approach adopted is geometric, using methods from
singularity theory.
