Brain-inspired spiking neural networks (SNNs) have demonstrated great potential for temporal signal processing. However, their performance in speech processing remains limited due to the lack of an effective auditory front-end. To address this limitation, we introduce Spiking-LEAF, a learnable auditory front-end meticulously designed for SNN-based speech processing. Spiking-LEAF combines a learnable filter bank with a novel two-compartment spiking neuron model called IHC-LIF. The IHC-LIF neurons draw inspiration from the structure of inner hair cells (IHC) and they leverage segregated dendritic and somatic compartments to effectively capture multi-scale temporal dynamics of speech signals. Additionally, the IHC-LIF neurons incorporate the lateral feedback mechanism along with spike regularization loss to enhance spike encoding efficiency. On keyword spotting and speaker identification tasks, the proposed Spiking-LEAF outperforms both SOTA spiking auditory front-ends and conventional real-valued acoustic features in terms of classification accuracy, noise robustness, and encoding efficiency.