If you were to publicly perform or record "Happy Birthday" or "Danny Boy" you would owe someone some money. Maybe. It's a little unclear.

According to Wikipedia, the melody of "Happy Birthday to You" was written in Louisville, Kentucky by sisters and teachers; Patty Hill and happy birthday song mp3 download
Mildred J. Hill in 1893.

You could just imagine the very morning they in greeted their classroom to a little tune entitled "Good Morning to All". Sadly, we don't know who wrote the lyrics to the song as we know it today, and so, the the heirs of the lyricist missing in action will never see their due!

The version as we know it was copyrighted in 1935 and was purchased by Warner Chappell in 1990 for US$15 million! With the value of "Happy Birthday" estimated at US $5 million, some musician and his/her heirs definitely missed out on a nice check.
While the current copyright status of the song is unclear, Warner claims that unauthorized public performances of the song are technically illegal unless royalties are paid to it.

In contrast, the mystery of the well known song Danny Boy has indeed been solved. Or has it? A very interesting read by Michael Robinson who reveals that the song Danny Boy is one of over 100 songs composed to the same tune.

According to Wikipedia, the root of the tune Danny Boy begins with a song titled Londonderry Air. Should Londonderry Air be the melody that carries over 100 songs, surely we should be giving huge credit to its composer. Unfortunately, this great melody and its original composer is still shrouded in a cloud of mystery.

What is known for certain, is that the lyrics of Danny Boy were originally written for a different tune in 1910 by an English lawyer, Frederick Weatherly. It's interesting to note that that the songwriter never actually visited Ireland, yet the tune would become a national anthem for the Irish.