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You can search pretty much whatever you want online -- searching for information is not a crime. But certain searches are monitored and certain words will trigger suspicion and investigations, and if you engage in illegal activity online, that is a crime.
You can search the words "kiddie porn" for example -- how else will you find information on the topic? -- but you absolutely cannot download the stuff. People can and do get arrested for their illegal online activities. But it's important to distinguish between suspicious searches and illegal activities. Googling the word "murder" does not make you a killer, and this principle extends to terror, porn, and more.
A Cautionary Tale
In 2013, a woman in New York wrote about how her family got a visit from police after certain searches emanating from her home's computers triggered terror-related suspicions. The words were used in disparate searches by different people. She looked for "pressure cookers" while her husband sought "backpacks" and her son read up on homemade bombs in articles linked to stories about the Boston Marathon terrorists.
It seemed as if the result of these unrelated actions -- when viewed together by experts trained to look for trouble online -- was that they looked linked. This led local authorities to suspect that someone in the home might be about to engage in terror. So Suffolk County Criminal Intelligence Detectives showed up at the front door.
Or so Catalanao believed. After her story got a lot of attention online, the police publicly clarified what really happened. In fact, the visit was not prompted by three random searches but by one single tip -- Catalano's husband's former employer was suspicious about his work searches called the cops. This prompted police to follow up. In fact, they said, the police would not otherwise have been interested in this family.
What to Avoid
Of course you should avoid trouble and you should be wary of potential problems that could come with search for some types of information. People who have claimed to be investigating child porn for artistic or journalistic purposes have been prosecuted, but that doesn't happen after one accidental link click. It takes much more.
Nor is there a set of magic words you could use that would make all your web searches seem innocent. Similarly, use of exclusively innocent language will not ensure that you are beyond suspicion. According to one blogger who fears her own crazy searches will get her arrested, even the words "pork" and "cloud" can be viewed in suspicion in some contexts.