Contra all the younger people who think that tradition automatically equals lace veils and mantillas, here’s a picture from 1965 of a typical bunch of American Catholic ladies dressed up for Mass. (They’re welcoming Pope Paul VI to New York in 1965, at St. Patrick’s Cathedral; but the level of formality is pretty typical of what I vaguely remember Catholic ladies wearing on Sunday in Ohio in the early 1970’s.) If you click on the photo to see it enlarged, you will notice a lot of nice hats and simple scarves on the female half of the congregation, but not many lace veils at all. Some women don’t seem to be wearing anything on their heads, but they may be wearing gauzy see-through scarves, which were pretty common. Here’s a smaller shot that gives more of a close-up when enlarged; you’ll see some winter hats too. Another crowd shot of the Pope leaving – you can see some faces here.
But before Vatican II, here’s a huge crowd of be-wimpled nuns and be-hatted Catholic ladies at the Budapest Eucharistic Congress in 1938. By Margaret Bourke-White, no less.
This is not to say that it’s wrong to wear veils to church. It’s just modelling another part of tradition than what was done in my part of the US, or in most chunks of Europe.