The current version 1.3 of the REST API is lagging behind. Currently there are more features available through the UI, and not everything is achievable through REST. We are planning to update the REST API and entirely redesign the documentation.

Available methods

For all available methods in the latest API refer to this page. Click on the methods to see examples.

The examples incorporate all available options/parameters parameters. To summarize:

The export options (/snapshots POST) are:

checkCustomFieldValues, includeAttachmentFiles

Regarding filters, boards, dashboards: for project snapshots you can only pass lists of their ids. For system you can choose to includeAllFilters, includeAllBoards, includeAllDashboards.
 

Import options (/deployments POST) are as follows:
 
For System snapshots:
mergeVersionsAndComponents, mergeProjectRoles, systemDoNotDeployGlobalPermissions
 
Project:
mergeVersionsAndComponents, mergeProjectRoles, createNewProject, newProjectKey, newProjectName
 
Project with issues:
mergeProjectRoles, createNewProject, newProjectKey, newProjectName, attachmentFilesPath
 
 
Skipping descriptions, avatars and reindexing, as well as selective merge and custom field resolution are not yet available through rest.

Hope this helps! Stay tuned for updates.