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The January 2005 issue of This Rock ("Quick Questions") dealt with the issue of veils:
Q: Did the Vatican ever publish a document stating that women are not supposed to wear head veils to church anymore?
A: No. Women are free to wear a head covering to church if they so desire. It’s just not required.
The document Inter Insigniores [ link ] by the Congregation for the Doctrine of the Faith (October 15, 1976) stated that the 1917 Code of Canon Law (canon 1262.2) requiring women to wear veils on their heads was a custom of the period and that such ordinances "concern scarcely more than disciplinary practices of minor importance." Thus the obligation "no longer has a normative value." But, as a sign of respect, women still are required to wear a veil when meeting the pope.
Here is the passage referred to above, from that document:
Another objection is based upon the transitory character that one claims to see today in some of the prescriptions of Saint Paul concerning women, and upon the difficulties that some aspects of his teaching raise in this regard. But it must be noted that these ordinances, probably inspired by the customs of the period, concern scarcely more than disciplinary practices of minor importance, such as the obligation imposed upon women to wear a veil on the head (1 Cor 11:2-6); such requirements no longer have a normative value.
Colin B. Donovan, STL, gave a reply on this question at EWTN ("Head Coverings in Church").
What little I've said about this in the past amounted to an urging of Catholic women to (by all means) wear a veil if they want to do so, but not to impose any such obligation on others, since the Church does not do so at this time, or act as if they are more obedient or spiritually superior in so doing. Nor should women who don't wear it frown upon those who do (assuming the latter don't exhibit questionable attitudes just described).
I think they're beautiful and graceful-looking myself (particularly ones like that pictured above): like bridal veils. Insofar as the intention is as a sign of modesty and femininity and submission to God, that's great. On the other hand, my wife has never worn one to church. Live and let live. Worship and let worship.