I can't see anything "anti-English" in the Fields of Athenry, and I've never met any English person who saw it that way. Rebelling against "the Crown" is something a lot of English people have felt called to do at various times, and have seen as their patriotic duty.

Maybe that's a quibble - there are songs which would qualify. But the relevance of making that quibble is that it's quite easy to identify songs as partisan, and they get used in a partisan way, when they aren't in themselves partisan at all. Singing a particular hymn can be seen, and can be intended as, a provocative and sectarian act - and then you find it's in a hymn book of the other religion anyway, used in another part of the world.

There's stuff which is intentionally and explicitly sectarian or racist or whatever, and anyone singing it except in a special kind of quasi-academic context, is setting out to make a sectarian or racist statement, and maybe laying down a territorial claim of some kind. And there is stuff which gets taken up and used as some kind of banner, when the content is in no way offensive ("I'm for Ever Blowing Bubbles" becomes a way of expressing support for West Ham, and could even get your head kicked in.)

It's easy enough to say "don't be offended" and if its a question of something that might be aimed at you or yours, that's probably something to aim for. But there's the question of how to react if it's aimed at someone else - I don't think there's anything particularly admirable in keeping quiet when some bigot is throwing insults at Black people or Jews or Arabs. Or Catholics, if you're a Protestant, or Protestants if you're a Catholic, if you're in a part of the world where those kinds of insults carry weight.