Beyond global search, the conversation list has its own controls for narrowing and ordering whatever sidebar filter you’re currently viewing.
Status
A dropdown lets you filter the list by conversation status, and you can select more than one at once:
Sort
Sorting by Oldest inside Unassigned is a fast way to make sure the conversations that have been waiting longest get picked up first, rather than always grabbing whatever just came in.
Going further
Combine Status and Sort with the Filter (funnel) icon for field-level conditions — see Advanced filters — and save any combination you use often as a saved inbox view instead of rebuilding it each time.