When you never, ever, ever, ever want to see it again, take my railgun system and shoot it off into space. Time it so that it avoids big star systems, and accelerate it faster than 11km/s, and it's gone forever.
Trash is a problem because we have it sitting around in landfills. The landfills smell. The landfills require a lot of work so as not to seep toxins into the water supply. The landfills attract unwanted animals and hobos that want to eat the decaying sandwich and pizza crusts. Some of the more toxic waste requires more elaborate procedures to keep it contained. But if it were far enough away in space, it wouldn't be anyone's problem ever again.