Are dogs allowed on public transportation?

I don't think dogs are allowed on public transportation in my area in California. If they are allowed (on buses, subway, etc.) they must be in a crate. Service dogs, of course, are allowed.

They are not allowed on most school playgrounds and parks in my area.

One thing really surprised me is that some stores do allow dogs. Some have no policy towards them. I have seen people bring in small dogs in purses and tote bags. The store clerks say nothing.