Supercaps have enough leakage to make them only useful as backup
during short periods, and are used sometimes to permit battery replacement
without losing state (The RTC in a digital camera for example could be
done this way).
Specialised high-efficiency supercaps are available for power use, but
will probably always be uneconomic compared to a battery for powering a
microcontroller - batteries store orders of magnitude more charge and
have almost constant voltage.