CDM simulations predict that there are hundreds of lumps of with masses greater than 10e7 solar masses in the Milky Way halo. However, we know of only a dozen dwarf satellites close to this mass. Are these lumps simply lacking in stars or is there a fundamental flaw in our most popular cosmology? By studying the tidal debris of known satellites we can potentially address this question. In this paper, we quantify the the effects of the dark matter lumps on tidal tails. The lumps scatter stars in the tidal tails from their original orbits producing a distinctive signature. We simulate debris evolution in smooth and lumpy halos potentials and use our simulations to motivate and test a statistical measure of the degree of scattering apparent in the angular position and radial velocity measurement of debris stars - the ``scattering index''. We find that the scattering index can in general distinguish between the levels of substructure predicted by CDM cosmologies and smooth Milky Way models, but that the sensitivity of the debris depends on the orientation of the parent satellite's orbit relative to the largest lumps orbits. We apply our results to the carbon star stream associated with the Sagittarius dwarf galaxy (Sgr) and find that these stars appear to be more scattered than we expect for debris orbiting in a smooth halo. However, the degree of scattering is entirely consistent with that expected due to the influence of the Large Magellanic Cloud, which is on an orbit that intersects Sgr's own. We conclude that the current data is unable to constrain CDM models. Nevertheless, our study suggests that future data sets of debris stars associated with other Milky Way satellites could provide strong constraints on CDM models.

20 pages, 7 figures, submitted to ApJ