To start off, the word "clairsentience" contains the words "claire" which means "clear" and "sentience" which means "feeling." This "clear feeling" people get is literally just that -- a feeling. Many people describe feeling sensations of clairsentience, even though they don't describe them as such. Anytime someone says they went with their gut, they are possibly experiencing a moment of clairsentience. As with all psychic abilities and everything paranormal, not everyone thinks clairsentience exists, and many people believe in it but keep their belief extremely private.
I see in the news all the time that people say "something didn't feel right." Maybe they felt like they needed to check on their child in the night, or that their spouse was in trouble during a trip. One of the passengers on the Malaysian Airlines Flight 370 gave his wife his watch and wedding ring before he departed on his trip, telling her to give them to their two young sons if something happened to him. There are numerous documented cases of soliders' dogs crying out suddenly as if they are in pain, the families realizing later that in that precise moment their child or sibling was killed. Police officers have described knowing that someone had a gun that was not visible, giving them just enough of an edge to doge a bullet and return fire without being injured.
I could go on, but I think everyone probably gets the point by now. I'm a big gut person. The times when I've mistrusted that strong feeling inside of me, things have turned out bad. When I trust the feeling, good things happen. Some of the best things in my life have happened as a result of trusting that gut feeling. Of course, as with other such psychic gifts, such feelings only work if you are truly in touch with yourself and the source of such feelings. when it comes to the source, there is certainly quite an array of beliefs out there.
Have you ever experienced a moment of clairsentience and either followed or not followed your feeling?