The preferred basis is (roughly) the eigenbasis for the interaction term between the thermal environment responsible for decoherence and the decohering system in question, since those are the states that are robust to thermal noise.
Outside the perturbative regime, there isn't a preferred basis because there aren't any classical observers.
To add to this, the states robust to entanglement with environment were dubbed “pointer states” by Zurek since they are the quantum states most closely analogous to the “pointer” of a measuring apparatus showing one particular result.
Outside the perturbative regime, there isn't a preferred basis because there aren't any classical observers.