Excavating into the shallow Martian subsurface has the potential to expose stratigraphic layers and mature regolith, which may hold a record of more ancient aqueous interactions than those expected under current Martian surface conditions. During the Spirit rover's exploration of Gusev crater, rover wheels were used to dig three trenches into the subsurface regolith down to 6--11 cm depth: Road Cut, the Big Hole, and The Boroughs. A high oxidation state of Fe and high concentrations of Mg, S, Cl, and Br were found in the subsurface regolith within the two trenches on the plains, between the Bonneville crater and the foot of Columbia Hills. Data analyses on the basis of geochemistry and mineralogy observations suggest the deposition of sulfate minerals within the subsurface regolith, mainly Mg-sulfates accompanied by minor Ca-sulfates and perhaps Fe-sulfates. An increase of Fe2O3, an excess of SiO2, and a minor decrease in the olivine proportion relative to surface materials are also inferred. Three hypotheses are proposed to explain the geochemical trends observed in trenches: (1) multiple episodes of acidic fluid infiltration, accompanied by in situ interaction with igneous minerals and salt deposition; (2) an open hydrologic system characterized by ion transportation in the fluid, subsequent evaporation of the fluid, and salt deposition; and (3) emplacement and mixing of impact ejecta of variable composition. While all three may have plausibly contributed to the current state of the subsurface regolith, the geochemical data are most consistent with ion transportation by fluids and salt deposition as a result of open-system hydrologic behavior. Although sulfates make up >20 wt.% of the regolith in the wall of The Boroughs trench, a higher hydrated sulfate than kieserite within The Boroughs or a greater abundance of sulfates elsewhere than is seen in The Boroughs wall regolith would be needed to hold the structural water indicated by the water-equivalent hydrogen concentration observed by the Gamma-Ray Spectrometer on Odyssey in the Gusev region.