On the Structure Board you can use the structure widget not only for showing structures, but also for searching existing issues (using JQL or text search) and displaying your clipboard contents.
To search existing issues:
- Make sure you have a structure open
- Click the structure's name in the main panel
- Look for the Tools section in the drop down
- Select either Text Search or JQL Query
Once the search is open, just start typing and the results will be updated.
Just like when you're working with a structure, you can select a specific view for your search results and then add and arrange columns as necessary.
Search only looks for issues from structure-enabled projects.