Lizzy, the only standard locations I've ever gotten something from a box or barrel in are No Man's Landing, the Racing Pits, and the Northern Slurries. Then the Lagoon and Fogstone Mines/Tunnels/Depths, but those have issues getting access. I've asked before if anyone has gotten anything other places, but nobody has corrected me so far.
Regarding frequency I've tried clicking every box and barrel in those locations multiple times per day. Northern Slurries only ever gives 1 item per day, and which one is random...including sometimes the barrels half hidden behind the Grease Pits entrance. I'm sad when it's only a solvent or ice flower, happy when it's special ice or special fur. I'm personally hoping that when Chapter 4 is released a new area will open up where things like green ice are a lot more frequent. Good luck.
p.s. Racing Pits usually gave me 2-3 items per day...magnets are good, but also some special metals.
p.p.s. No Man's Landing gives a bunch of items per day, so keep clicking barrels and boxes there on every trip through, because it's currently the only source of cloth.