Not illegal whatever your search. But, there are things which can make you appear of the "radar" of stalkers. If there was an issue, you could readily show that you were researching a paper. read more
You can search for anything, even how to make a bomb, etc. HOWEVER if you are ever charged with a crime or being looked at as a suspect in a crime then the police can get a court warrant to search your house and take your computer and then can use your past google searches against you in court. read more