When I have something that really needs a good cleaning I use laundry detergent and hot water.
I have cats and it has never bothered them.
Those scratchy green pads work great for tough scrubbing too.
Sorry I don't know of any commercial product that works well and is safe for cats too.
I do know that Lysol and Pinesol are bad for cats.

Now that I've gotten a good bit of the clutter cleaned up, I'm tackling appliances and surfaces that need serious attention. Ordinary household cleaners are giving me a struggle; they just won't do the job properly. Any suggestions as to cleaners that I could use that would be pet-safe?

I have two cats in the home. However, I will be moving with them in about a week into a clean environment. I am going to continue to clean this house with an eye on future rental or sale. But at that point, I won't need pet-safe cleaning materials anymore.