Most people prefer the use of aqueous electrolytes (eg. 1M H2SO4, 6M KOH, etc.). However, if you want high energy density of supercapacitor device you can try in organic electrolytes or ionic liquids. For using acidic or basic aqueous electrolytes you also have to see the stability of your materials in the electrolyte (for eg. metal oxides are unstable in acidic electrolytes). Another immportant parameter to understand is the pore size distribution of the material (in case of EDLC supercapacitor). The pore diameters should be adequate enough for quick transfer and adsorption of the electrolyte ions.
I would say the compatibility of the electrolyte with your electrode materials is the most important factor you need to consider. Like Namrata mentioned, some metal oxides, for example, ZnO, are readily soluble in both acidic and alkaline electrolytes. If you want to maximize your device's energy density, you might also need to choose electrolytes that allow wide potential windows.
A rule of thumb is to check the literature about your active materials or similar materials for any possible electrolytes.
I completely agree with Tianyu Liu and Namrata Deka . To add, it is advisable to test with aqueous electrolyte initially and then go for organic electrolytes as organic electrolytes are used commercially (requires high energy density) to see how your material fares with that of any material commercially established. Literature survey definitely helps you in this regard. The compatibility can be tested from its cycling performance. Good luck.
It depend on your active material. Aqueous electrolyte are the most used due to their low cost and high conductivity. Liquid ionic and organic elctrolyte can be used to extend the potential window but still suffer from the lower conductivity and toxic nature of organic electrolyte.
For my experiences i use only aqueous electrolyte with concentration of 0.5-1M.
Some of can be used to achieve higher capacitance ( but you will loss the cycle life of your device) and some are suitable to achieve excellent stability ( up to 30.000 ) but the capacitance still limited as comapred to others