Use query parameters (GET) for search. It's extremely useful to be able to share a URL to a search. (IE, Wikipedia and Google do this. Example: https://en.wikipedia.org/w/index.php?search=query+parmaters&... Example: https://www.google.com/search?q=query+parameters&oq=query+pa...)
Don't open new tabs when clicking links to narrow down a search. (To be quite honest, links should almost never open a new tab. Users who want to do this know how to do it and you don't need to make the choice for them.)