Like if I have a search auto suggestion script that says aXXo, when someone types ax...... or when they type Hancock I it would show a suggestion for Hancock 2008? Or a search cloud of popular torrents that inserts the search into the form like torrents.to does? Would I start getting DMCA notices?
That's an interesting question actually. If type in "Batman Begins T"... into google the suggested search entry will suggest to you "Batman Begins Torrent". As far as a know google has not had any trouble with this, which is a good indicator. Google does however filter results if it gets a DMCA notice and will remove the link from the SERPS. Simply suggesting what to search for also doesn't guarantee you will be taken to content that infringes on content, it could easily be a legitimately released torrent for the Batman Begins trailer. The bottom line is you should not have any trouble with having suggested terms that could eventually lead to copyrighted material. You must however comply with any DMCA notices you get to remove offending pages from the SERPS.
you might get hassled if you started ranking highly for the terms. Whether or not they had a legal case is debatable. In Sweden, where piratebay is based, there is no way to stop you 'pointing' to something else, i.e. linking, in the same way no one could stop you publicizing a phone number that had a recording giving away the 'secret' ending to Sixth sense and so on. Although they might be able to have the person running the phone line hauled up in court