Here's Fred Astaire doing "We Saw the Sea" from Follow the Fleet. I sing this all the time, but people just give me looks askance. Pshaw. It's funny.
And Fred and Ginger doing "Never Gonna Dance" from Top Hat.
Embedding disabled, but still, watch it: http://www.youtube.com/watch?v=wXT0e6HWndQ
Fred Astaire singing "Let's Face the Music and Dance." I love this song. And to think they originally said of FA "Can't act, can't sing, can dance a little." It's just the song, no video to speak of:
...And Ginger Rogers singing "Let Yourself Go," another one I often sing. It starts about 54 seconds in (the beginning is instrumental). Video is a slideshow of pictures from the dance of the same title, but really you're better off just watching them dance. Unfortunately there's no singing though in that version.
Some days I wish I had tv solely so I could watch Turner Classic Movies. I miss that.