Choose two people to leave the room - these two people won't need to do any singing (everyone else will), and will be competing against each other. Once they've left, each person needs to pair up with another person, and come up with a song between the two of them. Each person in the pair then chooses a subsequent line of that song. For example, if a pair chose the song "You Are My Sunshine", one person would have "you are my sunshine, my only sunshine", and the other would have "you make me happy when skies are gray". This would be the lines of the song that these two people would sing. You now repeat this process to find a second line of a different song with a different pair. At this point, each person should have two lines, from two different songs. Now each person chooses a hand for each line. For example, I might choose my left hand to be "You make me happy when skies are gray", and my right hand to be the other line that I chose with the other partner. When that's done, the two people who left come back in. They now have to match up each song with its other line, in the correct order. This is done as follows: The person in the middle will tap the hand of a person, and they will sing the line that that hand corresponds to. For example, if they tap my left hand, I would sing "You make me happy when skies are gray". Each person has a go, which consists of tapping two hands sequentially, and listening to the song sung in response. If you tap two hands of the same song in the correct order, you get a point and those two hands are put behind the people's back who sung them (removing them from the game). I.e. if you tap the hand of the person with "You are my sunshine, my only sunshine", and then you tap the hand of the person with "You make me happy when skies are gray", you get a point and then those hands are removed from the game. The two people in the middle are competing against each other, alternating their goes, and the person at the end with the most points once all the pairs have been found wins.
Make sure people feel confident enough to sing very short bits in their group first. I guess you could do this with humming, although it'd be very hard!