If you want to save a contact search on the People Search page for future use, click on the Recent Searches tab.

Click on the Recent Searches title (if the pane beneath is closed).

Then recover the relevant search you want to save by the date and time of the search you had conducted.

In case you want to view all of the filters included in that search, click on the small arrow in the relevant search box to expand the view.

To save this search for future use, click on Save in the relevant search box, rename this search, and click on Save.

Now you can easily access this search in the Saved Searches tab by clicking on the Search button next to this specific search: 

You can edit the name of the search, share it with your team members or remove this search by clicking on the Menu icon on the right side of this search (3 dots)