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The quick search is the fastest way to define search criteria. However, it is less precise than other search methods for complex queries (e.g. project = Jira AND status = Open AND priority = High).
It is most useful when your search criteria is not complex, for example, you know the project key and some key words for an issue.

To use the quick search: Enter your search criteria in the search box in the header bar of Jira and press Enter.
Tip: If you know the issue key or project key, enter it before other search terms, e.g. "JRA help link is broken".  

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More detailed information can be found on the following Atlassian page: Quick searching 

The basic search is more precise than the quick search, but easier to use than the advanced search.
It provides a user-friendly interface that lets you define complex queries, without needing to know how to use JQL (advanced searching).

To use the basic search: Navigate to Issues (in header) > Search for issues, then enter your search criteria.
Tip: If the advanced search is shown instead of the basic search, click Basic next to the Image Modified icon.
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More detailed information can be found on the following Atlassian page: Basic Search 

The advanced search is the most powerful of the three search methods. You can specify criteria that cannot be defined in the other searches (e.g. ORDER BY clause).
However, you need to know how to construct structured queries using the Jira Query Language (JQL) to use this feature.

To use the advanced search: Navigate to Issues (in header) > Search for issues, then enter your search criteria.
 Tip: If the basic search is shown instead of the advanced search, click Advanced next to the Image Modified icon.
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You can choose which details are shown about the results, by choosing the Columns that should be displayed.

More detailed information can be found on the following Atlassian page Advanced search in Jira