Drupal - How can I search for a specific user on the People administration page?

That's all you can do with core Drupal - it is a little frustrating for sure. :-)

However, there are lots of contrib modules that let you do more, such as

• Advanced User, in which user fields to filter/search by can be configured in the module's settings or
• the simpler User search to People Admin which simply puts the user search box on there for you.

Of course, you can also opt to make yourself a user-based view and create an admin page yourself by using relationships and filters. :-)

Hope one of these helps!

Assuming this question is about D7, only using Drupal core wont be sufficient. However, you could create a view (using the Views module) of all your users, and use the views style provided by the dataTables module. Here is a quote from the dataTables project page:

The DataTables module integrates the jQuery plugin DataTables into Drupal as a views style and a callable theme function. DataTables lets you add dynamic features to tables, including:

• Variable length pagination
• On-the-fly filtering
• Sorting with data type detection
• Smart handling of column widths
• Themeable by CSS
• And more ...

You'd be especially interested in the On-the-fly filtering. For way more details about this module, refer to "What are my options to visualise a views table with many fields grouped in columns?".