A user with the Create'Saved'Search right has access to RT's saved search system. Without this right, the "Create a saved search" box is hidden within RT's search interface.
In RT 3.4, I also needed to grant the ShowSavedSearch right for the box to show up.
This right can be granted to users and groups on a global level only.
Need to enable ModifySelf for user to allow "My saved searches".