I'm fortunate in that Seren never really seemed to want to go digging through the kitchen trash.
The bathroom trash if there were tissues in it? She loves shredding tissues.
One of the things that I've always done to keep dogs out of the kitchen trash, though, is that I never put wet, smelly stuff in the trash. Anything like that goes into a container in either the refrigerator or the freezer, and it only goes in the trash on trash day right before I take the bags out.
It helps that I have twice a week trash collection at my house.