Improves the search for users in the backend significantly: Search for first name, last, email and more of users instead of only nicename.

Lava Ajax Live Search Provides Ajax live search for posts, custom post type in your WordPress site. It also works for BuddyPress, BBPress search

See what your users are searching for. Sort the results by keyword or search count

Users Searched For is a lightweight tool that records all searches made on your WordPress site. Detects if results were found.

The plugin adds an advanced search form with search suggestions using a shortcode.