I'm convinced that everyone who plays the piano should sing while at the piano. I had a piano teacher once who assigned me a Mozart Sonata to work on. When I practiced it and came back to play it,he stopped me after a few bars and said "sing that opening phrase for me". He could tell I'd learned the notes of that phrase,but not the nuances of its phrasing-which notes are more accented,how they flow,etc.-the stuff that makes that music come to life. My vocal rendering of the phrase proved his point. He sung the phrase the way it was supposed to sound,and we started working on the piece in this way.
When working on melodies,chords,bass lines,rhythms,whatever your doing at the piano,singing that melody,a note in the chord [or the notes of the chord stacked]that bass line or rhythm helps you to HEAR it better. When you can really hear what you are to play [as opposed to just finding the notes on the keyboard] helps you to PLAY it better-to make it come to life. Singing songs allows you to accompany yourself. Many of us don't like the sound of our singing (and/or speaking) voice, or feel that we simply "can't sing". I say hogwash. to quote an old Sesame Street song ( "Sing a Song") - "don't worry that it's not good enough for anyone else to hear-just sing.....sing a song". Singing will make you a better pianist,and add to your enjoyment of piano and music in general. Happy humming, Will