I have a variety of cats, some are pretty calm, one I'm sure would sit happily in an open basket as he wants to know what I'm doing at all times, he is a total loon and runs behind the bike when I'm trying to go out but one of my girls who is soft as muck went totally feral in the cat carrier and has not been to the vet in a number of years since she broke out of the lovely wicker basket one, wearing an e collar and slashing my hand as I tried to undo the door so she could get out without strangling herself, whether she knew that the vet was a rip off sadist or she had some kind of claustrophobia I'm not sure but lucky we were still in the house and so she's never been since.
The rest of the beasts are generally calmer in a cat box but I'm thinking to get one of those proper top loading metal cage carriers and secure that to a front bike carrier. Some of the more relaxed cats would I'm sure be ok on the rear pannier rack, I have a friend that does that, just uses really secure bungees so its easy on and off and all hers are fine.
I have also got a cargo trailer and have a storage issue, poor bike has to live outside but the trailers nearly all fold fairly flat and the wheels come off so storage is not too bad, I did think to use it in case of need for the vets but don't feel comfortable having the cats so far out of sight and at risk from traffic and would go for the rear rack if there was an emergency any time soon.
Do watch out for the hitch attachment with the trailers, I'd avoid any trailers with clamp hitches they are too easy to work loose, I read a lot about them before I got mine warning of that, I got mine off gumtree and it did have a clamp and could see where it had been along the road, I bought an axle fitting and swapped them over fairly easy. Check the specs re room on your axle, quick release wheels and some kinds of brakes are not suitable. I've only needed to use it once so far as I can pile huge amounts of 'stuff' on the bike itself but its good to have just in case I need to move furniture or trees.