The previous owner had installed a pull-out cabinet trash can under the sink, which we thought it was really nice since the trash can is not visible in the kitchen, but it was a two-step process... first you would have to open the cabinet door then pull out the trash can...
Obviously, you would have to push back in the trash can then close the door once you are done using the trash can.
The more modern kitchens have a built-in pull-out cabinet trash can that is attached to the cabinet door so that when you open/pull the cabinet door, the trash can becomes immediately available for use then you can just close the cabinet door once you are done!
I've been thinking about converting my cabinet trash can for a while then finally purchased the parts from Lowe's and installed it.
This is what it looks like and it comes with a trash can. If you want to attach this to the cabinet door then you also need to buy a separate item that is specifically designed for attaching to a cabinet door. Surprisingly, these things are not cheap but I found this particular model on sale.
I removed the cabinet door off its hinges then carefully drilled two very thin plywoods on it so that I can permanently attach the pull-out trash can rail onto the plywood by drills instead of drilling onto the cabinet door, which has the risks of damaging the cabinet door.
I'm very happy with the final product. It glides out so smoothly and easily.
If you are thinking about doing this project, make sure you measure the height and depth of your cabinet under the sink because they have many different sizes available. I would also recommend sturdy stainless steel model instead of cheaper white aluminum model.