Vernier onset/offset thresholds were measured both psychophysically and with the steady-state VEP by introducing a series of horizontal breaks in a vertical square-wave luminance grating. Several diagnostic tests indicated that the first harmonic component of the evoked response generated by periodic modulation of offset gratings taps mechanisms that encode the relative position of spatial features. In the first test, a first harmonic component was only found with targets that contained transitions between collinear and noncollinear states. VEP vernier onset/offset thresholds obtained with foveal viewing were in the range of 15–22 arc sec. Control experiments with transitions between symmetrical, noncollinear patterns (relative motion) did not produce first harmonic components, nor did full-field motion of a collinear grating. A second series of experiments showed that VEP thresholds based on the first harmonic component of the vernier onset/offset response had an eccentricity dependence that was very similar to that found in a psychophysical discrimination task that required a left/right position judgment (vernier acuity). Other recordings showed that the first harmonic of the vernier onset/offset VEP was degraded by the introduction of a gap between stimulus elements, as is the displacement threshold. The vernier onset/offset target also produced a second harmonic component that was virtually identical to the one produced by a relative motion stimulus. Displacement thresholds based on these second harmonic components showed a more gradual decline with retinal eccentricity than did the first harmonic component elicited by vernier offsets. The second harmonic of the vernier onset/offset VEP was relatively unaffected by the introduction of gaps between the stimulus elements. The first and second harmonic components of the vernier onset/offset VEP thus tap different mechanisms, both of which support displacement thresholds that are finer than the resolution limits set by the spacing of the photoreceptors (hyperacuity).