Self-organized polymer patterns resulting from the evaporation of an organic solvent drop on a soluble layer of polymer are investigated. The patterns can be modulated by changing the rate of evaporation and also the rate of substrate dissolution controlled by its solubility. Both of these affect the contact zone motion and its instabilities, leading to spatially variable rates of substrate etching and redeposition that result from a complex interplay of several factors such as Rayleigh-Benard cells, thermocapillary flow, solutal Marangoni flow, flow due to differential evaporation, osmotic-pressure-induced flow, and contact-line pinning-depinning events. The most complex novel pattern, observed at relatively low rates of evaporation, medium solubility, and without macroscopic contact-line stick-slip, consists of a regularly undulating ring made up of a bundle of parallel spaghetti-like threads or striations and radially oriented fingerlike ridges. Increased rate of evaporation obliterates the polymer threads, producing more densely packed fingers and widely separated multiple rings due to a frequent macroscopic pinning-depinning of the contact line. Near-equilibrium conditions such as slow evaporation or increased solubility of the substrate engender a wider and less undulating single ring.