I've had cats since I was 10, and aside from the fact that they really do love sleeping in moses baskets/car seats/piles of freshly wahed baby clothes and blankets, I've never had a problem with them. I have been a little paranoid about them getting on top of babies in the night, but then they are not allowed to sleep in our room or the baby's room so it's not really a problem.
The only problem that I did have was with my dog when I was pregnant with my youngest, and that was more to do with her breed (Border Collie) and the fact that I was the centre of her universe. She was a rescue case (in my opinion) that I had bought from a woman who lived in a shoebox sized flat in Peterlee and it became clear to me over the course of the first couple of days that she had been mistreated. The first walk that we went out on, I said 'wait' as a car approached and she flattened herself to the floor, looking panicked, and was always frightened of men. Anyway, to cut a long story short, as I got more and more pregnant I found it difficult to take her out as much as I had done, so DH and kids did it, and this upset her greatly. She became really clingy and started snapping at my youngest, amongst other things, really trying to be in the middle of everything all the time. We decided the kindest thing to do was to re-home her and, after a month of vetting people, we found a lovely woman who was absolutely besotted with her animals and Mollie went to live with her. That was almost five years ago, and we still keep in touch. It was absolutely the kindest thing we could have done for her and the problems were as a result of her personality and her breed (super intelligent), rather than the fact that she was a dog. I'm pretty sure that if I'd had a slightly less bright dog, it wouldn't have been a problem at all.