Joyce,  I hope this helps a bit in your search for the children's song
"Horace The Horse."  The Harry Fox Agency's list of all the songs for
which it is the collecting agent for mechanical licenses is always
a good place to start a search if you don't know a lot about a tune.
The URL for the agency, explaining its purpose and scope, is

        http://www.nmpa.org/hfa.html

The URL for its search engine, allowing you to search by title key
words or by composer, is

        http://www.nmpa.org/search_form.html

Of course the agency has no interest in public domain material, but
your tune sounded like it was new enough not to be in the public domain
yet.  Sure enough, this listing turns up as being in the agency's
repertoire: "song code: HO5288.  Horace the Horse (On the Merry-Go-
Round).  Cy Coben, Charles Grean."

I tried searching the Bagaduce Music Lending Library to see if they
had the sheet music, but came up dry.  However, I'm sure the tune was
deposited for copyright with the Copyright Office, Library of Congress.

But for much sheet music registered in recent decades, since nobody
buys sheet music any more, all that gets deposited to secure copyright
registration is a lead sheet for the tune.  This is particularly true
for tunes that go directly onto phonograph record or movie sound
tracks.  But as a last resort, try the U.S. Copyright Office.

Matthew Caulfield