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Non-canonical phenomena - defined here as observables which are either insufficiently characterized by existing theory, or otherwise represent inconsistencies with prior observations - are of burgeoning interest in the field of astrophysics, particularly due to their relevance as potential signs of past and/or extant life in the universe (e.g. off-nominal spectroscopic data from exoplanets). However, an inherent challenge in investigating such phenomena is that, by definition, they do not conform to existing predictions, thereby making it difficult to constrain search parameters and develop an associated falsifiable hypothesis. In this Expert Recommendation, the authors evaluate the suitability of two different approaches - conventional parameterized investigation (wherein experimental design is tailored to optimally test a focused, explicitly parameterized hypothesis of interest) and the alternative approach of anomaly searches (wherein broad-spectrum observational data is collected with the aim of searching for potential anomalies across a wide array of metrics) - in terms of their efficacy in achieving scientific objectives in this context. The authors provide guidelines on the appropriate use-cases for each paradigm, and contextualize the discussion through its applications to the interdisciplinary field of technosignatures (a discipline at the intersection of astrophysics and astrobiology), which essentially specializes in searching for non-canonical astrophysical phenomena.