I have a soft spot in my heart for the sentimental songs of the 19th century. Here are some books I'd recommend:

• Heart Songs (National Magazine & Chapple Publishing Co., 1909)
  The magazine surveyed its readers for suggestions for a collection fo the "popular melodies of today and the days gone by"). The book is available as a $40 paperback reprint from a genealogy publishing house, but I find copies on the original in used book stores for under $10.
  The magazine also published two volumes of sentimental poetry titled Heart Throbs - the poetry books don't seem to be as easy to find as the songbook.
• Read 'Em and Weep (1926 & 1945) and Weep Some More, My Lady (1927), by Sigmund Spaeth.
  These are very interesting and entertaining studies of the old, sappy classics. Spaeth also wrote an handy book called Barber Shop Ballads and How to Sing Them, and a number of other books on old popular music and on classical music.
• Songs That Never Grow Old (syndicate Publishing Co., 1909)
  This is a nice collection of familiar songs. There are some interesting photos of singers in the first several pages. I wouldn't say this is an "essential" book, but it's nice. Readily available for under $20.
• Lost Chords: The Diverting Story of American Popular Songs (by Douglas Gilbert, Doubleday Dorand and Co., 1942)
  Gilbert isn't as entertaining as Spaeth, but he tells some fascinating stories about the songs in his book.
• Song Dex Treasury of Humorous and Nostalgic Songs (Song Dex, Inc., 1956)
  This is an early fake book, designed, I suppose for electronic organs. It has 740 songs, enough to make you cry a big bucket of tears.
• My Pious Friends and Drunken Companions: Songs and Ballads of Conviviality (Frank Shay, 1927)
  I guess you'd call these songs "barroom ballads." They're certainly entertaining, although I wish the book had tunes for more of the songs. There's a More Pious Friends sequel, and both books are also available bound in a single volume.

I'd say the Spaeth books and Heart Songs and maybe Pious Friends are the "essential" books in this category.
-Joe Offer-