I would like to search JIRA's "Quick Search" from Chrome's Omnibox. This is not the same as this Chrome Omnibox search string:
https://myserver/jira/browse/%s
That string will only open perfectly (not partially) matched JIRA IDs. A Quick Search will automatically open the issue that uniquely matches the search criteria--even if the search criteria is a partial match. For example, consider a system where there is only one JIRA issue that contains -77, and that JIRA issue is CLS-77. Using Quick Search (at the upper-right corner of the JIRA site) to search for "77" will open issue CLS-77 automatically. Performing the same search through Chrome Omnibox custom search string I listed earlier will not launch CLS-77 when searching for 77.
In newer versions of JIRA (I'm using 6.2.7), the URL to use is https://jira.example.com/secure/QuickSearch.jspa?searchString=%s (replace jira.example.com with your host name).
Update: In JIRA 6.4.4, this URL works: https://jira.example.com/issues/?jql=%s