Does anybody know of any tricks to make it more likely that your villagers will give up their sloppy furniture? I've seen a few of my villagers have these items and I'm checking everyday in the police station and recyclable shop and have never seen any!

As far as I know there's not much you can do .. I do believe that if you send villagers with it in their homes similar sized items they might replace it and sell, but I don't think even that's guaranteed

The only real trick is to have a town full of cranky villagers. They seem to be the ones most likely to seel, though I have had snooties put up sloppies for sale at the flea market. After my recent cranky moved, I stopped getting them completely.

I get pieces all the time in my side town.  Just removed all the items from retail and waited.  
Removed anything they put up for sale and one day a sloppy clock was there.  Also found full sets of cardboard furniture.  It's a town where I don't speak to the villagers and don't care what their houses look like.   There are 5 villagers that put items up in that town.

In my main town I found a sloppy clock that Tutu put up for sale.  That's the only item I've found there.

I did the same thing - bought everything in Retail and then waited. I have never received sloppy furniture through the mail from villagers (I only have one town), but I have bought most of it through Retail.

Nothing worked for me. Gave em a table & sloppy table disappeared, never went to retail. Bought set for 10 million then realized I don't like the sloppy look and put it another characters house.

I made a laundry room in my house in the back room.  I keep sloppy furniture in there because of all the clothes usually slung onto the pieces. Thought it went well with a laundry theme