Context: Container-based virtualization is gaining popularity in different domains, as it supports continuous development and improves the efficiency and reliability of run-time environments. Problem: Different techniques are proposed for monitoring the security of containers. However, there are no guidelines supporting the selection of suitable techniques for the tasks at hand. Objective: We aim to support the selection and design of techniques for monitoring container-based virtualization environments. Approach:: First, we review the literature and identify techniques for monitoring containerized environments. Second, we classify these techniques according to a set of categories, such as technical characteristic, applicability, effectiveness, and evaluation. We further detail the pros and cons that are associated with each of the identified techniques. Result: As a result, we present CONSERVE, a multi-dimensional decision support framework for an informed and optimal selection of a suitable set of container monitoring techniques to be implemented in different application domains. Evaluation: A mix of eighteen researchers and practitioners evaluated the ease of use, understandability, usefulness, efficiency, applicability, and completeness of the framework. The evaluation shows a high level of interest, and points out to potential benefits. © 2021 The Authors