Transverse 2D phase space distributions of a 2.1 MeV, 5 mA H- beam are measured at the Proton Improvement Plan II Injector Test accelerator at Fermilab with an Allison scanner. This paper describes the design, calibration, and performance of the scanner along with the main results from 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 under different beamline conditions. The amplitude 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 amplitude in the beam core is found to be decreasing approximately exponentially with action and to be phase-independent. Outside of the core, the amplitudes decrease with action at a slower rate than in the core. This "tail" comprises 10%-30% of the beam, with 0.1% of the total measured intensity extending beyond action 10-20 times larger than the rms emittance. The transition from the core to the tail is accompanied by the appearance of two "branches" that are separated in phase and extend beyond the core. A set of selected measurements shows that there is no measurable emittance dilution along the beamline; the beam parameters are practically constant over a 0.5 ms pulse; and scraping in various parts of the beamline is an effective way to decrease the transverse tails by removing the branches.