Actually, the shower is the one place where I don't sing. I don't multi-task very well.
But I do sing everywhere else and highly recommend it. I usually don't purposefully sing--it's a spontaneous outburst that accompanies an unexpected moment of happiness (Doesn't that have a name?). I'm often surprised this occurs as I go about my daily routines, especially in the dead of winter when there's little to be cheerful about.
I feel very fortunate that I can sing (bass, by the way). I was taught to read music at an early age, sang in my mother's church choirs (she was a music major), high school and college choirs. Since I don't attend church anymore, I rather miss singing in the choirs.
Singing is a great outlet, a stress reliever of sorts. There's something wonderful about letting it all out in a loud, strong musical voice. I make up a lot of songs and have written many in my mind. I think I could become rich if I had them published. George Gershwin Smith. But it will never happen, so I'll just have to remain satisfied with my own self's "odes to joy".