Yes to foxes (urban foxes are a huge thing in London), yes to rats, and yes to them ripping open and pawing through the rubbish. It's disgusting. But people still do it. And the fly problem in summer is unbelievable.
We have to have bear-proof containers, and while they now sort of work, we all still wait to put them out until right before the trashtruck comes.
My first bear-proof container wasn't, and I retain a fond memory of the bear sitting at the end of the driveway, legs extended and spread in front, grabbing the can, pulling it to its chest, wrapping its hind legs around the can to stabilize it, and using its front legs to bear-hug it until the lid went "spang!" and popped into the air. I went out righteously with a slingshot and strode up the driveway to the bear. I told him indignantly that that was My Garbage, and directed him to leave. But he did not leave. He stuck his head into the metal can, clear in. So I whanged a ball bearing into the bottom of the can, and it made a great noise, a little reminiscent of the 'cannon shots' in the 1812 overture. Then he did leave, and left me thinking, "Fredbear, you self-righteous geezer, you are going OD in property rights. Did you even hear yourself? Asserting a fee simple interest in garbage. The bear is hyperphagic. He needs that garbage more than you or Western Disposal need it." (This is fallacious, of course, but it convinced me not to be pompous about garbage raiding, except insofar as it is a gateway drug to dog-killing, which is very common around here, though the mountain lions are more at fault than the bears.)