I think every standard tenancy contract/advert states no pets, the same as it quotes no sharers and no benefits most of the time.
I've lived in 4 different rental houses now, (not blocks of flats which can be more difficult), and have not had any problems so far. All the places have been let unfurnished, so its my stuff Max destroys
, and I have a clause to steam clean all the carpets, dry clean the curtains, and the standard deposit lodged with the agents in case of any damage beyond standard wear and tear.
I've done a lot of searching to find the right places for Max to live, being an indoor cat we've always insisted on stairs for him to get exercise.
It is possible, but I guess it depends on where you are looking and budget as well.