We investigate the dewetting of aqueous, evaporating polymer [poly(acrylic acid)] solutions cast on glassy hydrophobic (polystyrene) substrates. As in ordinary dewetting, the evaporating films initially break up through the nucleation of holes that perforate the film, but the rapidly growing holes become unstable and form nonequilibrium patterns resembling fingering patterns that arise when injecting air into a liquid between two closely spaced plates (Hele-Shaw patterns). This is natural because the formation of holes in thin films is similar to air injection into a polymer film where the thermodynamic driving force of dewetting is the analogue of the applied pressure in the flow measurement. The patterns formed in the rapidly dewetting and evaporating polymer films become frozen into a stable glassy state after most of the solvent (water) has evaporated, leaving stationary patterns that can be examined by atomic force microscopy and optical microscopy. Similar patterns have been observed in water films evaporating from mica substrates, block copolymer films, and modest hole fingering has also been found in the dewetting of dry polymer films. From these varied observations, we expect this dewetting-induced fingering instability to occur generally when the dewetting rate and film viscosity are sufficiently large.