I just read about how people get mesmerized by a singing dummy, so it's good to work a song into a routine. Although it takes some practice, I think this is true: if you can sing, you can vent-sing. If you can't sing, it's not going to sound any better with your lips not moving.
The only barrier might be that when you're vent-talking, you can stay away from words with definite Bs, Ps, Ms, etc. and all the other plosives and their kin. It is a little more tricky to do a popular song and not sing words that people are anticipating. Then again, an audience anticipating the word might make them unwittingly fill in a sound in their head that they didn't actually hear.
So doing Madonna singing Like an Irgin, or Judy Garland's Nerry Little Christnas might be a real crowd pleaser.
(Note, I deliberately walked away from all the "singing dummy" jokes. Felt too easy to name a bunch of them, and I haven't decided whether rap should be called singing...)