It isn't a practical thing to send anything down to the ocean floor to specifically look for garbage down there. The depths are too great, as would the expenses. But we know for a fact that, over the years, we have dumped an amazing amount of crap into the ocean and then walked away whistling all out-of-sight-out-of-mind. The problem is we don't know just how much.
Researchers at the Monterey Bay Aquarium Research Institute in California knew one way you could look down there, though. Just use the footage from the expeditions that have been down to the ocean floor already. They're just one aquarium, based in Monterey, and so they just focused on Monterey Canyon, which sits between Monterey and Santa Cruz, but they had 22 years of footage to go off of, ranging from January 1989-January 2011. That was about 18,000 hours of footage, and from that, they noted (this is a link to the actual study) 1,537 bits of trash. 33% of the trash was plastic (and 54% of that was plastic bags), and 23% was metal (of which 67% were cans). 14% was rope. After that was unidentified debris (7%; anything that didn't fit into the other categories, like mop heads, wound up here, as well as anything they couldn't figure out what it was), glass (6%), fishing debris (5%), paper (4%), other fabric (3%), rubber (2%), clothing (1%), and sub-1% amounts of manufactured wood, abandoned research equipment, ship wreckage, military, concrete and batteries. At shallower depths, a lot of it was rope and fishing debris; plastic and metal tumbled down to the bottom of canyon slopes and peaked at the mid-range depths, and things got less identifiable (or less categorizable), wound up at deeper depths.
1,537 pieces of garbage may not sound like a lot, but then, remember that they were taking footage from expeditions that weren't even looking for it, that they didn't have the technology to get down to the depths with the bulk of the trash until about halfway through the time period, and it was nowhere close to a complete scan; they only had footage from 0.24% of the survey area. A straight extrapolation- not going to give an accurate figure, but it's worth giving a ballpark idea- would make it 640,417 bits of trash over the entire Monterey Canyon region. To say nothing, of course, of all the undersea regions on Earth that aren't Monterey Canyon.
Here's a video if you need a visual.
Also, note that the trash on the seabed will keep for longer than it would on the ground. The water is cold, there's not much sunlight, and there's not much oxygen, meaning bacteria that would break it down won't grow as well. Once a piece of trash gets colonized, it's best off staying down there- think of all those shipwrecks that get overgrown with barnacles and coral and such- but the best solution is to not let it get down there in the first place.
In conclusion, do be mindful of your crap. If a storm drain says it drains to the lake or the ocean, this is what happens to it.