I think that there is no list because it is a two parameter problem. The different LCs have different polarities and for this reason different solution behavior. For example 5CB (4-pentyl-4'-cyanbiphenyl) is low to medium polar so that it can solve medium polar polymers like SAN (24% or better lower acrylonitrile content) or benzylmethacrylate. MMBA, p-methoxybenzyliden-p-butylanilin is slightly more polar, so that SAN (24% or more acrylonitril content) or polycaprolactone, probably also PMMA, polylactide or polypropylenecarbonate could be solved in it. PAA, p-azoxyanisol, which is nematic between 116 and 135°C, is very polar so that polylactide or PMMA are suitable polymers. Cholesterylbenzoate (nematic between 146 and 179°C) is very unpolar so that polystyrene or polycyclohexylmethacrylate could be solved in it. An interesting high Tg polymer which is soluble in unpolar solvents like toluene as well as in polar solvents like ethylacetate is APEC, an aromatic polycarbonate from Covestro. It should be soluble in a wide range of LCs.
Adding a tiny amount of polymer can change a fluid dramatically, with drag reduction, change in turbulence and flow instability, and possibly no-slip boundaries as examples (for example, http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.83.5591). There might be an industry application where speeding-up the processing of something is quite important. According to Billmeyer (Textbook of Polymer Science), dissolving polymers is a slow process of "solvation-unfolding-swelling". The information you gave is very useful. Initially, I guessed that liquid-crystalline polymers (not cross-linked) were good candidates.
So, from my understanding, polarity is very important. Other that that, it is more of a "trial-and-error" approach, rather than making a good guess by looking at the solubility parameter or chemical characteristics of a possible solvent.
You typed "drug" instead of "drag". How misleading that is. You are right drag reduction is an old topic. But there are still many things to do with adding polymers in complex liquids.