Machine Unlearning (MU) algorithms have become increasingly critical due to the imperative adherence to data privacy regulations. The primary objective of MU is to erase the influence of specific data samples on a given model without the need to retrain it from scratch. Accordingly, existing methods focus on maximizing user privacy protection. However, there are different degrees of privacy regulations for each real-world web-based application. Exploring the full spectrum of trade-offs between privacy, model utility, and runtime efficiency is critical for practical unlearning scenarios. Furthermore, designing the MU algorithm with simple control of the aforementioned trade-off is desirable but challenging due to the inherent complex interaction. To address the challenges, we present Controllable Machine Unlearning (ConMU), a novel framework designed to facilitate the calibration of MU. The ConMU framework contains three integral modules: an important data selection module that reconciles the runtime efficiency and model generalization, a progressive Gaussian mechanism module that balances privacy and model generalization, and an unlearning proxy that controls the trade-offs between privacy and runtime efficiency. Comprehensive experiments on various benchmark datasets have demonstrated the robust adaptability of our control mechanism and its superiority over established unlearning methods. ConMU explores the full spectrum of the Privacy-Utility-Efficiency trade-off and allows practitioners to account for different real-world regulations. Source code available at: https://github.com/guangyaodou/ConMU.