The ever-growing power consumption of wireless communication systems necessitates more energy-efficient algorithms. This paper introduces SpikACom ({Spik}ing {A}daptive {Com}munication), a neuromorphic computing-based framework for power-intensive wireless communication tasks. SpikACom leverages brain-inspired spiking neural networks (SNNs) for efficient signal processing. It is designed for dynamic wireless environments, helping to mitigate catastrophic forgetting and facilitate adaptation to new circumstances. Moreover, SpikACom is customizable, allowing flexibly integration of domain knowledge to enhance it interpretability and efficacy. We validate its performance on fundamental wireless communication tasks, including task-oriented semantic communication, multiple-input multiple-output (MIMO) beamforming, and orthogonal frequency-division multiplexing (OFDM) channel estimation. The simulation results show that SpikACom significantly reduces power consumption while matching or exceeding the performance of conventional algorithms. This study highlights the potential of SNNs for enabling greener and smarter wireless communication systems.