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Singing in the shower is an idle habit of many hygiene-concerned persons. I sometimes sing in the shower (most of the time I'm just silent and almost asleep), and I know exactly why every time.

Most people just can't sit still. They need to be doing something. People might flip their pen, drum on their desk, scratch their face, or fix their hair. The reason why there is so much singing in the shower is that we have no pen, no desk, a face that's being scratched by water, and hair that's usually slicked back (or however you have your hair in the shower). What's left that still allows us to still clean our bodies? Singing.

It's just something to do.