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Deep audio classification, traditionally cast as training a deep neural network on top of mel-filterbanks in a supervised fashion, has recently benefited from two independent lines of work. The first one explores "learnable frontends", i.e., neural modules that produce a learnable time-frequency representation, to overcome limitations of fixed features. The second one uses self-supervised learning to leverage unprecedented scales of pre-training data. In this work, we study the feasibility of combining both approaches, i.e., pre-training learnable frontend jointly with the main architecture for downstream classification. First, we show that pretraining two previously proposed frontends (SincNet and LEAF) on Audioset drastically improves linear-probe performance over fixed mel-filterbanks, suggesting that learnable time-frequency representations can benefit self-supervised pre-training even more than supervised training. Surprisingly, randomly initialized learnable filterbanks outperform mel-scaled initialization in the self-supervised setting, a counter-intuitive result that questions the appropriateness of strong priors when designing learnable filters. Through exploratory analysis of the learned frontend components, we uncover crucial differences in properties of these frontends when used in a supervised and self-supervised setting, especially the affinity of self-supervised filters to diverge significantly from the mel-scale to model a broader range of frequencies.