This is one of a series of questions submitted by attendees of the Copyright, Common Mistakes and Myths Webinar and answered by Laura N. Gasaway, author of Pocket Copyright Guide for Publishers.
Answer: As with quoting other material, the amount quoted makes a big difference. Lots of old ballads have very long lyrics while pop music has many fewer with much repetition. So, assume that it is only a line or two from a long ballad. That is likely a fair use. But if it is a few lines from a pop tune, then permission is likely required. Typically, publishers require authors to get the permission, although some publishers offer assistance with this.
Pocket Copyright Guide for Publishers by Laura N. Gasaway and edited by Iris Hanney contains information vital to the publishing community.