Don't get me started on this!
It's not just that I'm a composer myself, but I think the music scene is all the poorer for our collective failure to see writers as artists. In the classical world it was always the composer who got the credit, and I think this attitude started once performers - (in the music hall days?) started paying writers to produce material. Gradually this became industrialised till we arrived at the Brill Building et al, and songs became product more than art (with occasional revolts, of course).
I think it may be significant that this state of play already existed when the collectors started in ernest. In the absence of known composers - and with a mass of anon material in public ownership, no-one gave any thought to the writer-artist. And of course there was no 'star' either, so attribution (and even lionisation) was affixed to the song-carrier and/or source singer, while the crucial fact that there have always been great craft writers and re-writer/arrangers from all walks of life was largely forgotten.
So the problem of writers not getting legally-due royalties is only the tip of the (cultural) iceberg.
That said, the PRS and other collecting agencies are quite good, if slow, at working out what is in copyright and eventually getting the money to the right person - whatever the performers, promoters, consumers and generally ill-informed or downright ignorant (like those hymn-fakers) might say.
When MySpace started, I had a serious go at persuading the PRS to insist that there should always be a field in the upload system for 'composer' - and you couldn't complete the process without filling it in. Ok, people would have put the singers name in there too, but at least they'd have had to pause for thought, and it would have prevented the problem Genie mentions above - because both the 'best-known' performer and the perhaps guessed writer would be right there on the screen.
Sadly, PRS had bigger fish to fry with MySpace, and I got nowhere.
Now, of course, up and downloading is a massive phenomenon, and the writer's rights have been swept away entirely.
If, like me, you also write papers and books and scripts etc, then songwriting is no different - I'm a writer, who happens to include music in my oeuvre. I fail to see why my compositions should be treated with any less respect than, say, my academic work - but sadly not everyone agrees - specially a sector of the folk world who have grown up believing that all songs belong to everyone and always have.
I'm for ever banging on about the need for accurate attribution, even for works in the public domain - because it's the best (only) way to keep composing up there with sculpture, playwriting, painting etc.
I'm just now finishing a commissioned musical made entirely out of traditional/PD songs and tunes. I've written new words to most of them, and adapted the rest. So as well as the script and song book there is a third book - which outlines where I found the tunes, what the original words were (at least of the version I mainly had in mind when I started bodging), what we know about the songs (other titles / tunes etc) and websites etc for further info.
PS Of course: Paul Anka only wrote new lyrics for My Way. As Wki nearly says: It was based on the French song "Comme d'Habitude" composed in 1967 by Claude François and Jacques Revaux, with lyrics by Claude François and Gilles Thibault. Anka's English lyrics are unrelated to the original French song so he's credited and has copyright (with permission of the composers) as the author.
(In law it's composer for tunes, author for lyrics).