I am trying to find an aprotic ionic liquid which is immiscible with certain solvents such as DMF, DMSF, THF, NMP etc. What are the criteria I should be checking in selecting ionic liquids?
Try looking into fluorinated anions and long alkyl chains in the cation. They are usually not soluble in water and many polar solvents, but they are not very environmentally friendly.
The easiest thing to do is to get a bunch of plausible solvents to try and check for yourself. This is relatively quick and oftentimes more efficient than spending all your time looking for some information that may be very difficult to find (or may not even exist).