The organic solvents are regarded as a good solvents to dissolve most polymer such as benzene, Toluene, Xylene..... some types of polymer can be dissolved in water or ethanol.
While it depends on the individual repeating units of the monomer, trying NMP (n-methyl pyrrolidone) or xylene or toluene could be tried. Remember these are carcinogens. Whether the polymer was formed using addition vs. condensation polymerization also makes a difference what solvent you can use. In general it is easier to dissolve a polymer synthesized using condensation polymerization such as PMMA using common solvents.
actually, it is depending on the repeating units that form the polymer chains, and what kinds and how many Functional groups on these chains, but in general for synthesis traditional polymer benzene, Toluene, Xylene are public to use, for natural polymers there are other solvents
Well if the discussion is about synthetic polymer, then i agree with the response of Sandeep Chandrashekharappa . DMF and DMSO are the strongest solvent.