Idioms are figurative expressions whose meanings often cannot be inferred from their individual words, making them difficult to process computationally and posing challenges for human experimental studies. This survey reviews datasets developed in psycholinguistics and computational linguistics for studying idioms, focusing on their content, form, and intended use. Psycholinguistic resources typically contain normed ratings along dimensions such as familiarity, transparency, and compositionality, while computational datasets support tasks like idiomaticity detection/classification, paraphrasing, and cross-lingual modeling. We present trends in annotation practices, coverage, and task framing across 53 datasets. Although recent efforts expanded language coverage and task diversity, there seems to be no relation yet between psycholinguistic and computational research on idioms.