Oh boy, I sure wish I'd known about the private screenings. I've been ok with my own kitties going through TSA, but my tabby girl tore a friend to shreds who was transporting her for me.
We use Sherpa brand bags that can be carried three ways: as a shoulder bag, as a backpack, or as a handbag. I'm very happy with them and have used them for cats and dogs (my dogs are cat-sized).
An incredible number of dog show and cat show folks use the
SturdiBag. It was designed by a flight attendant and really nice. I was just about to buy some of those when I encountered the Sherpas on a very good sale and ended up with them.
There are off brands too and they are probably fine, but personally I recommend Sherpa or SturdiBag.