I think I was shopping at a store when there was a dog that urinated on my purse that was sitting in the cart. Throughout the night, (not sure if it was part of the same dream), I think I also dreamt of possibly cats who may also have been urinating.

Any thoughts?

Dreaming of an animal urinating on your purse sounds like you feel that somebody could be ruining something that you value in waking life.

Since dogs and cats are supposed to be trained not to do that sort of thing, it could be that you feel that this is happening because somebody is not behaving properly.