Search Reports are divided into the same two types as other reports. "My Saved" are the reports you have created and "Shared" are the reports others have shared with you. Mouse over each report type to see your options for each report. You have the option to download, edit, or delete each Saved report and download or hide your Shared Reports.
Dashboards aren't really a report. They are a list of Saved and Shared Dashboards that have accumulated over time. Use the mouse over option to delete each Saved report and hide your Shared Reports.