Natural light fields are threaded by lines of darkness. For monochromatic light, the phenomenon is familiar in laser speckle, i.e., the black points that appear in the scattered light. These black points are optical vortices that extend as lines throughout the volume of the field. We establish by numerical simulations, supported by experiments, that these vortex lines have the fractal properties of a Brownian random walk. Approximately 73% of the lines percolate through the optical beam, the remainder forming closed loops. Our statistical results are similar to those of vortices in random discrete lattice models of cosmic strings, implying that the statistics of singularities in random optical fields exhibit universal behavior.