How can I search for a specific user on the People administration page? I can filter by Role, Permission and status on the people page, is there an easy way to search for one specific user?
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!