As a Catholic I know many Catholics who are both Monarchists and Republicans. To be honest most English Catholics tend to be Monarchists or pro-Monarchy and most Irish, Welsh and Scottish Catholics tend to be Republican or pro-Republican in their sympathies.
Of course when one travels to France or Spain, many Catholics (but not all) tend to have Monarchist sympathies.
I have mixed emotions, but none of them very strong. As a Welshman I tend to feel our Monarchy was robbed from us back in 1282, with a brief interlude by Owain Glyndwr. On the UK level my sympathies are (were!) very Jacobite, and I am aware that the British monarchy 'wuz robbed' from Catholics by the tricks and twists of the protestants via the Dutch and the Germans with dubious links to what was a Catholic English legitimate Royalty.
Add to this that the British Monarchy has been used as a substitute religion (worship of the UK state) to replace the Catholicism ripped from our people 500 years ago which had a monarch that was always subject to the moral guidance of Holy Mother Church.
The monarchy today is still a closed shop to Catholics, that and many of the hangers-on get huge handouts whilst setting a poor example to the public at large.
Still, given that the dearth of a monarch may lead to President Blair or President Cameron, sometimes I feel it's a case of better the devil you know...
So, in short I would say that I believe in having a monarch, albeit that the monarch should be able to be a Catholic, and that the extended amount of freeloaders should be curtailed. Whether that monarch should be the head of Wales as well as England and Scotland... well, I have mixed feelings.
I also feel the monarch has let us all down by green-lighting all sorts of dubious laws to do with Faith and Morals (esp on abortion and homosexuality).
But I'd be fascinated to hear what other Catholics feel. If you have the time also please vote in the poll placed in the right-hand column.
P.S. Before anyone says it, yes I do know that Heaven is an absolute monarchy; but surely that is because we know God is always benevolent, all-loving and does no wrong. Here on earth, on the other hand, our society falls far short of these standards given man's fallen nature and the murky world of politics.