I've never been very keen on singarounds that just go round in a circle taking turns. Reminds me of courses I've attended where you start by going round with everyone introducing themselves - the "Creeping Death" as someone described it.

One thing I tried a couple of times was to take a pack of cards, shuffle them, give each singer a card at the start of the round, and take them in order. The randomness gives the session a spontaneity to get the ball rolling.