Fr. Zuhlsdorf has a lively discussion going on regarding head coverings for women and has a poll asking what you think about it. Visit and vote! The comments in the combox are very interesting.
When I returned to the Catholic Church in 2008, I didn't intend on falling in love with the Traditional Latin Mass, but I did. The first time I attended, I was surprised to see how many women wore a mantilla (The lacy veil women wear in Spain.). Within a week, I found a veil at my local Catholic bookstore and bought it. That Sunday, I wore it and haven't attended our Extraordinary Form Mass without it.
There are a few reasons why I love the veil. Years before I returned to the Church, I was intrigued with Jewish prayer shawls. The tradition of men taking the prayer shawls and covering their head with it as they prayed, touched me deeply. I was reminded of Moses who went before God and covered his face to protect the sacred result of that meeting. Covering one's head signified reverence and an appreciation for the sacred.
I bought a prayer shawl from a woman who had visited Israel and brought some back. I remember at times using it as I prayed and really liked it. At the time, I had no idea about women's head coverings apart from the Amish. But I liked feeling as though I was entering into a sacred space for prayer and having something over my head accomplished that purpose.
The first time I wore the veil to Mass, I had that same feeling. I also knew of the verses from 1 Corinthians 11:3-7:
But I want you to understand that the head of every man is Christ, the head of a woman is her husband, and the head of Christ is God. Any man who prays or prophesies with his head covered dishonors his head, but any woman who prays or prophesies with her head unveiled dishonors her head--it is the same as if her head were shaven. For if a woman will not veil herself, then she should cut off her hair; but if it is disgraceful for a woman to be shorn or shaven, let her wear a veil. For a man ought not to cover his head, since he is the image and glory of God; but woman is the glory of man.
Wearing a head covering signified an understanding that there is a hierarchy of submission when we pray to God. What some may not understand is that Scripture says both men and women need to submit, not just the woman. The woman has an outward sign of that submission.
For me personally, I like it. I don't think it should be required because if it were, feminists would have a field day denouncing the Church and men for being controlling. I think what is happening now is that more women are starting to examine this practice and experience for themselves the way it affects their heart during worship. That to me is a very good thing.
For those who aren't sure, I'd say try it. Some are starting to veil for the Novus Ordo Mass, which is a little different, only because the tradition of wearing a head covering is not common at all. (In fact, I rarely see women wearing even a hat.) I've worn my mantilla at times within a Novus Ordo, and other times a hat. I know whenever I've done it, I've always felt as though I've entered a secret place with God. But that's me.
I'll be curious to see how Fr. Z's poll turns out. Remember to vote and check the comments if you have time!