Transverse 2D phase space distribution of a 2.1 MeV, 5 mA H$^-$ beam is measured at the PIPIT test accelerator at Fermilab with an Allison scanner. The paper describes the design, calibration, and performance of the scanner as well as the main results of the beam measurements. Analyses of the recorded phase portraits are performed primarily in action-phase coordinates; the stability of the action under linear optics makes it easier to compare measurements taken with different beamline conditions, e.g. in various locations. The intensity of a single measured point (pixel) is proportional to the phase density in the corresponding portion of the beam. When the Twiss parameters are calculated using only the high-phase density part of the beam, the pixel intensity in the beam core is found to be decreasing exponentially with action and to be phase-independent. Outside of the core, the intensities decrease with action at a significantly slower rate than in the core. This `tail comprises 10-30% of the beam, with 0.1% of the total measured intensity extending beyond the action 10-20 times larger than the rms emittance. The transition from the core to the tail is accompanied by the appearance of a strong phase dependence, which is characterized in action-phase coordinates by two `branches extending beyond the core. A set of selected measurements shows, in part, that there is no measurable emittance dilution along the beam line in the main portion of the beam; the beam parameters are practically constant over a 0.5 ms pulse; and scraping in various parts of the beam line is an effective way to decrease the transverse tails by removing the branches.
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