Dyes that can stain oil and water
The diazo dye (1), known as Solvent Yellow 124 or Somalia Yellow is soluble in aliphatic hydrocarbons. In the European Community, it is used to mark low-taxed diesel fuel (for heating purposes). In order to prevent its use in cars, samples are acidified. This results in acetal cleavage and protonation of the diazo-fragment, yielding the red, water-soluble iminoquinone species (2)



Since this is probably not exactly what you wanted, what about dyes that exist in a neutral and a zwitterionic form, depending on the polarity of the solvent?



Brooker's merocyanine is one of the most prominent examples and show strong solvatochromism, i.e. the colour changes with the polarity of the solvent