I think it differs for everyone. I am reminded of Ricky Fitts from American Beauty who found extreme beauty and profundity in a plastic bag flying in the wind. For me, I will sometimes get a feeling that a certain moment is special and profound, and it usually happens when I am at a decisive moment in my life.

I remember when I decided to switch majors once while watching some ducks play around in the river. One duck was behind its friends, so I created a metaphor that it didn't matter where these ducks were along the long, winding river of life because their position in the river is arbitrary. So even if it took me extra time to complete school, my happiness was more important. I had another profound moment the other month when I randomly stumbled on some arrays of solar panels and realized that working in clean energy is the right path for me.

Being well-timed is an important condition for profundity in my opinion. We can walk through an art museum filled with art that is extremely deep and applicable to the human condition but not feel moved  by any of it and feel more emotion when looking at a child smiling with their parent. I actually had this exact experience in a museum in France; I found the tourists there far more interesting than the actual artwork because they were real people with real concerns versus inanimate artwork.

However, sometimes the art speaks to us more depending on the time in our life. It seems to me that anything can be profound in the right context.