REST recommends that queries (not resource creation) be done through the GET method. In some cases, the query data is too large or structured in a way that makes it difficult to put in a URL, and to address this, a RESTful API is modified to support queries with bodies.
It seems that the convention for RESTful queries that require bodies is to use POST. Here are a few examples:
- Dropbox API
- ElasticSearch
- O'Reilley's Restful Web Services Cookbook
Queries don't modify the internal state of the system, but POST doesn't support idempotent operations. However, PUT is idempotent. Why don't RESTful APIs use PUT with a body instead of POST for queries that require a body?
NOTE: A popular question asks which (PUT vs POST) is preferred for creating a resource. This question asks why PUT is not used for queries that require bodies.