Suffice, for starters, to say that I have found convincing the following analysis by a canon lawyer on whether the practice is still legally binding:
The Truth Unveiled: Head Covering Still Obligatory for Women Attending Mass
Apologist Patrick Madrid has also found this argument convincing and adds the helpful comment: "The fact that 'nobody does this anymore' is not a good reason not to observe this venerable Catholic custom." (The fact is, though, that this practice has not been universally abandoned, Deo gratias.)
But why does it matter? Surely there are more important things to discuss. I understand the position of those who don't consider this a very big deal. But here's why I consider this to be a sort of "paradigm issue".
We know that there are Traditions with a big "T" that are binding on all Catholics, that must be held in order to be a Catholic in good standing. And then there are traditions with a little "t", the various practices and customs which express, foster, and uphold our faith in myriad ways.
During the revolution of the past decades, two major things happened. First, many Catholics became convinced that, because they are at least in principle mutable, the traditions (small "t") could be changed willy-nilly. This turned out to be naive, I think, but I'm sure there were many individuals who went down this path in good faith. A second more sinister occurrence is that certain individuals and groups knew perfectly well that changing certain practices, small-t traditions, would actually change the faith of the people.
A good example of this is Communion in the hand. Yes, it was practiced in the early Church, so it's not intrinsically wrong. But in every Catholic Rite, from East to West, it had been discarded centuries ago as a practice fraught with practical and doctrinal difficulties. It was resurrected by the Protestant revolutionaries precisely in order to undermine faith in the Real Presence—they knew that some little-t traditions are pretty tightly coupled to the big-T Traditions they support and express. And then during the post-V2 liturgical revolution it was resurrected once again, not by the faithful, but by modernist prelates and groups like Call to Action. Do we have to wonder as to their motives?
It seems that in the aftermath of the liturgical revolution and now well into the counter-revolutionary phase, we have come better to understand the crucial role small-t traditions play in passing on the big-T Traditions in their integrity.
The veiling of women during the sacred liturgy has a much more venerable traditional pedigree even than Communion on the tongue, having been explicitly commanded by the Apostle and practiced universally from East to West until into the twentieth century. And what were the societal factors that were pushing for women to remove their head coverings? Were faithful groups like Catholics United for the Faith agitating for this, or was it not rather groups like the National Organization for Women with a very different agenda?
There are many Catholic truths expressed by this beautiful tradition, but one of them surely is the importance of gender distinction in God's created order—certainly that would seem to be expressed in St. Paul’s teaching in 1 Cor 11. Gender confusion is one of the greatest ideological challenges of our time. At the very moment when the prevailing culture was clamoring to flatten all gender distinctions, in the Catholic Church we saw the (illicit) abandonment of a major liturgical expression of that very truth. And with nothing put in its place to fill the void.
At least when Latin Rite Catholics in various countries were told to not to kneel anymore to receive Holy Communion, the injunction (albeit not very frequently obeyed) to approach the Sacrament with a profound bow was put in place to try and counter-balance the lost sign of reverence. But when women doffed the veil, what practice was put in its place to continue to express the Church’s teaching?
For these and other reasons, I see this as a perhaps small issue which nevertheless represents something much, much larger.
Perhaps that's precisely why this notion raises so many hackles, even among certain faithful Catholics. But I would ask one thing of those who are shocked, shocked I say at the notion that it might still be binding for women to cover their heads at Mass. Were you by any chance one of those Catholics who thought it was ridiculous for some of us to argue that the traditional Roman Rite, the "extraordinary Form" of the Roman Rite had never been legally abolished with the coming of the Novus Ordo? There were plenty of folks who did. Authors Kenneth Whitehead and James Likoudis laid out page after page in their book The Pope, the Council, and the Mass as to exactly how the traditional Rite had definitely been legally abolished—they considered it a certainty. And there was a certain plausibility to their arguments—it did indeed appear to be so. But there were nuances and principles involved that kept some of us from going along with the argument and continuing to insist that, despite certain appearances, the traditional Rite had not been legally abolished. Now, it turns out, they were wrong and we were right.
So, perhaps it would be worthwhile to keep an open mind about other potential examples.
A few counter-arguments have been offered to the idea that this tradition is still binding. Probably the most common one is a passage in Inter Insigniores. Here is the answer given by the individual who posted the canonical study to which I linked above, specifically treating the matter of this document:
Good, I was hoping someone would bring up "Inter Insigniores", from which your first point comes. There are several reasons why that little clause does not apply:
1. The direct and immediate object (or the "holding of the case", from a legal perspective) of that document was to affirm that only men could be admitted to the priesthood. The statement by Cardinal Seper on head coverings is obiter dicta, not essential to the holding and not binding as a pronouncement of law in any way. If this first point sounds overly legal to you, you shouldn't belong to a Church with a two millenia [sic] old tradition of canon law. Laws mean things, and rules matter.
2. The Cardinal was referring, not to women covering their head in church, but merely to the custom of women covering their hair everywhere, as had formerly in some parts of the world been the case. Read his exact words. There is nothing that compels the conclusion that he was referring to liturgical veiling. To say otherwise would be to say that the Cardinal intentionally made a somewhat seditious statement-- as this document came out before the 1983 Code and there was no doubt in anyone's mind that the Canon 1262 was binding.
3. This document was issued by the CDF, which does not have competence over liturgical law. If this document was designed to amend the Code of Canon Law of 1917, it would have to had come from the Pope himself. If it was designed to change liturgical law, it would have come from the congregation with the comptetence [sic] to do so.
(NOTE: there are two types of approval a congregation's documents can receive from a Pope: in colloquial english general and specific. Specific approval [forma specifica] is necessary for the document to be binding with papal authority. Inter Insignores was of the first kind-- general. Summorum Pontificum was of the second kind-- forma specifica)
(in the comments to "Patrick Madrid Weighs in on the Veiling Debate")
I think those are cogent reasons demonstrating that that passage of Inter Insigniores was not intended to, nor could it, overthrow a canon in the 1917 code—even if it was addressing the liturgical practice, which is in doubt, it lacked both the competence and the authority to do so. And the argument of the author of the canonical study demonstrates, I think, that even after the promulgation of the 1983 Code the practice of veiling stands both as a liturgical law and as an immemorial custom.
Certainly, the strongest argument against the practice still being binding is the complete lack of enforcement or even (more mildly) re-enforcement of the practice from the hierarchy. Strange to me that women are still expected to veil in the presence of the Pope, but not in the presence of our Lord. Still, it is my hope that there will soon be signals from Rome that, like kneeling for Holy Communion, this practice is indeed to be fostered anew.
In the end, though, I agree with those who have stated that this beautiful and venerable tradition will return by the voluntary practice of Catholic women, not on the basis of ecclesiastical legislation. It is happening and that is a very beautiful thing.