We propose trace pursuit for model-free variable selection under the sufficient dimension reduction paradigm. Two distinct algorithms are proposed: stepwise trace pursuit and forward trace pursuit. Stepwise trace pursuit achieves selection consistency with fixed p. Forward trace pursuit can serve as an initial screening step to speed up the computation in the case of ultrahigh dimensionality. The screening consistency property of forward trace pursuit based on sliced inverse regression is established. Finite sample performances of trace pursuit and other model-free variable selection methods are compared through numerical studies．