  Directional detection of WIMPs, in which the energies and directions of the
recoiling nuclei are measured, currently presents the only prospect for probing
the local velocity distribution of Galactic dark matter. We investigate the
extent to which future directional detectors would be capable of probing dark
matter substructure in the form of streams. We analyse the signal expected from
a Sagittarius-like stream and also explore the full parameter space of stream
speed, direction, dispersion and density. Using a combination of non-parametric
directional statistics, a profile likelihood ratio test and Bayesian parameter
inference we find that within acceptable exposure times (O(10) kg yr for cross
sections just below the current exclusion limits) future directional detectors
will be sensitive to a wide range of stream velocities and densities. We also
examine and discuss the importance of the energy window of the detector.
