Sophie and her chin

Seems as if Sophie Grace has developed feline acne. My husband thinks I'm insane. Her chin basically just looks like it's dirty, which is quite obvious with her white fur there. Upon closer inspection, it seems to be an oily substance (I scraped my fingernail acrossed it and it came away with a film of yuck).
I washed it with a bit of dish soap, but don't want to irritate the area. Upon searching here, I've seen everything from hydrogen peroxide to Stridex sensitive skin pads.
The girls both eat from ceramic bowls, washed daily. I have no idea how she got this, but hopefully, we'll get it to clear up soon. If not, off to the vet with her!

I have problems with 2 of my cats and they are the ones who love their wet food. I've had luck with the Stridex handling the more advanced cases. To reduce outbreaks, I use shallower dishes for their wet food and wash their chins with a damp washcloths when it looks like they are having problems. They still have breakouts but don't have it so bad that the vet needs to clean them out.