Abandoned mines? Old sinkholes?
Well landfills are basically holes. The problem with natural cavities is that chemicals will leak out and enter the groundwater. Landfills have liners to prevent this.
Some landfills might have liners which might prevent this.
I suspect that the holes may not be near where the rubbish is, and that filling old mines with rotting matter might well cause problems of its own - methane can be nasty stuff.
With a hole, you have to dig a hole, throw in trash, and then put layers of dirt over the trash to keep it from stinking up the area.
With a landfill, you do the same thing, except without going to the trouble of digging the hole first.
The Grand Canyon would hold a whole load of crap.
Just saying.
The archaeological evidence of throwing trash in a hole is abundant …
Used to be the common method: you’d dig a pit and dump your garbage into it. Eventually the pit filled, so you covered it with dirt and dug another one.
Of course, there was far less garbage back then. It consisted mostly of spoiled food.
We need a trash planet like in the movie Soldier, except we don’t have craft large enough to make the journey. As well as facilities established on another planet (obviously). This is what should happen in the distant future.
Phaw … recycle and compost … if you’re smart about your buying habits you can reduce your trash output by up to 75% …
Or they’d put the garbage in a pit, and then burn it. Doesn’t always work out so well.
Some places do use holes. Others just don’t have enough/big enough holes. And digging a hole to put trash in it costs money. Plus there’s the liner thing.
What’s the matter with building Mr. Trashmore’s? One in our area is grass covered except for the part with solar panels. Looks nice. Another is being reforested naturally.
The number one issue for landfills is getting the dirt to cover each day’s pile. And it can’t be any old dirt.
Even a country as large as the USA will eventually run out of places to dump garbage. Of course, the UK, with a much more concentrated population, has reached that point already and there are many different schemes to reduce or even eliminate landfill waste.
Waste is a major problem around the world and the scenario in WALL-E is looking less far fetched these days.
Like this?
The answers are: they did, they do, and sometimes they did new holes just for that purpose.
But using mines or whatever isn’t great because of contamination and shipping cost associated with getting the trash to the landfill.
How many abandoned mines and sinkholes do you think there are, just lying around ready to fill with trash?
I know a couple of decades ago there were all these news stories about garbage barges with nowhere to go. Ah, we’re running out of landfill space! Except that was all about a dispute between New York and the particular landfill they were using. The problem wasn’t landfill space, it was just some hardball negotiation tactics.
So we’ve got plenty of places to dump our trash, so we don’t have to go looking for abandoned mineshafts. WALL-E was not a documentary.
Better picture of a landfill under construction.
Paging Snowboarder Bo
Not for long – there’s a big river at the bottom, which would carry the trash away somewhere else. (That river is what made the canyon in the first place, after all.)
It would carry the trash to Mexico. Then it’s Not Our Problem! 
Probably already addressed, but there would be several problems. One is that a lot of old mines are full of water. Once the pumps are turned off they tend to fill up. Sinkholes would probably not be a good idea because of what caused them. Also, I’d think ground water contamination would be a serious issue for just tossing in general trash.