The primary observable in dark matter direct detection is the spectrum of scattering events. We simulate multiple positive direct detection signals (on germanium, xenon, and argon targets) to explore the extent to which the underlying particle physics, manifested in the momentum dependence of the operator mediating the scattering, can be extracted. Taking into account realization (Poisson) noise, a single target nucleus with 300 events has limited power to discriminate operators with momentum dependence differing by q^\pm2 for a wide range of dark matter masses from 10 GeV to 1 TeV. With the inclusion of multiple targets (or a factor of several more events on a single target), the discrimination of operators with different momentum dependence becomes very strong at the 95% C.L. for dark matter candidates of mass 50 GeV and above. On the other hand, operator discrimination remains poor for 10 GeV candidates until multiple experiments each collect 1000 or more events.

32 pages, 14 figures; references updated; revised to match journal version