Thanks for reminding me I was supposed to do some research on this, Lucy. At a music camp in January, I saw a wonderful play-party book from World Around Songs, 20 Colberts Creek Road, Burnsville, NC 28714, (704)675-5343. This thread reminded me I had planned to write for a catalog, which I did just now.
The best-known singing game book is, naturally, The Singing Game, by Iona and Peter Opie. It's out of print, but several other Opie books have come back into print lately, so I'm waiting rather than paying an exorbitant price at a used book store.
There are a couple in the Wee Sing" series that might help - I'd recommend Wee Sing Silly Songs and Wee Sing and Play, both by Pamela Conn Beall and Susan Hagen Nipp, published by Price Stern Sloan - about three bucks apiece. Tom Glazer used to have a number of books of singing games and the like, but they're all out of print.
Dover has reprinted a couple of classics: Games and Songs of American Children (William Wells Newell, 1903 - $9.95); and Old Songs and Singing Games (Richard Chase, 1938 - $2.95).
Check out this site (click) for an interesting bibliography, and also don't forget to search the world-renowned Digital Tradition for @playparty.
Somebody above suggested the books by Ruth Crawford Seeger, American Folk Songs for Children & Animal Folk Songs for Children. they're great books, and Mike & Peggy Seeger made recordings of all the sog songs in these two books their mother published. The CD's are available on the Rounder label from Dick Greenhaus at http://www.camsco.com - they're a great (and reasonably-priced) resource.