This Primer provides an overview of a fundamental set of analysis methods for studying waves, vibrations and related oscillatory phenomena — including instabilities, turbulence and shocks — across diverse scientific fields. These phenomena are ubiquitous, from astrophysics to complex systems in terrestrial environments, and understanding them requires careful selection of techniques. Misapplication of analysis tools can introduce misleading results. In this Primer, the fundamental principles of various wave analysis methods are first reviewed, along with adaptations to address complexities such as nonlinear, non-stationary and transient signal behaviour. These techniques are applied to identical synthetic datasets to provide a quantitative comparison of their strengths and limitations. Details are provided to help select the most appropriate analysis tools based on specific data characteristics and scientific goals, promoting reliable interpretations and ensuring reproducibility. Additionally, the Primer highlights best ethical practices for data deposition and the importance of open-code sharing. Finally, the broad applications of these techniques are explored in various research fields, current challenges in wave analysis are discussed, and an outlook on future directions is provided, with an emphasis on potential transformative discoveries that could be made by optimizing and developing cutting-edge analysis methods.