The Great Tune Thread is a useful listing, but not all the tunes are in the Public Domain. A quick glance spotted several, including "The Ballad of the Green Berets" (Sadler and Moore).
Even www.pdinfo.com has a disclaimer; the recent changes by Congress have muddied the waters.

Of course you are safe with anything in the Bodleian Library website or Bruce Olson's website, American Memory, pdmusic.org and nearly all of Contemplator- as long as you steer clear of specific arrangements (even of Greensleeves and esp. Scarborough Fair) that have been copyyrighted.

What you are asking for, an accurate, comprehensive listing, would require research on a large scale if 20th century music after ca. 1910 (1923 is no longer a valid cut-off for some renewed material) is involved. You might look up some of the threads on copyright here in this website.