Bichromatic optical Lissajous fields generated via second harmonic generation are described for a variety of Gaussian input fields that contain vector or elliptic singularities. The Lissajous figures repetitively traced out by the electric vector E during each optical cycle are more complicated than the familiar polarization ellipse of monochromatic light and can contain 0-3 self intersections. Eight different characterizations of these figures are discussed, some of which contain unexpected singularities. Six of these characterizations are expressed in terms experimentally measurable bichromatic Stokes parameters. As is the case for the monochromatic polarization ellipse, Lissajous figures can be assigned a handedness (right/left) that depends on the sense in which E traces out the figure. Two different forms of handedness reversal are discussed: these lead to two different Lissajous line singularities. Like the polarization ellipse, Lissajous figures can also be assigned an orientation that leads to a variety of point singularities. An input beam containing a single on axis vector (elliptic) point singularity with absolute charge n is found to produce an output beam that contains 12n+1 (6n+1) Lissajous point singularities.
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