It is widely believed that the perovskite Sr2RuO4 is an unconventional superconductor with broken time-reversal symmetry. It has been predicted that superconductors with broken time-reversal symmetry should have spontaneously generated supercurrents at edges and domain walls. We have done careful imaging of the magnetic fields above Sr2RuO4 single crystals using scanning Hall bar and superconducting quantum interference device microscopies, and see no evidence for such spontaneously generated supercurrents. We use the results from our magnetic imaging to place upper limits on the spontaneously generated supercurrents at edges and domain walls as a function of domain size. For a single domain, this upper limit is below the predicted signal by 2 orders of magnitude. We speculate on the causes and implications of the lack of large spontaneous supercurrents in this very interesting superconducting system.