Are you disgusted when you smash a rock and find a piece of Junk Ore instead of a gem? Do you become impatient with all the weeds that spring up constantly on every plot of land during Spring, Summer and Autumn? Are you disappointed when your fishing efforts net an old boot, a fishbone or an old tin instead of a Big Fish, Pirate Treasure or Fossil of Fish? Don't be! Every item in Harvest Moon can be utilised in a positive fashion. Every piece of rubbish has some value.
Obviously, you can ship any item of rubbish that you find but its shipping value is fairly minimal at 1G.
Old tins and fishbones are an acceptable gift for the Witch Princess. Her AP will increase by 300 points and her FP by 3 points when you give her a fishbone or tin. Do NOT make the mistake of giving her an old boot, though. She is quite particular when it comes to the rubbish that she values.
Apart from this, however, weeds, junk ore and rubbish can be extremely useful. In Spring, Summer and Autumn, collect and store weeds for cooking. There are a number of recipes that require a weed as an ingredient. You may make some of them only once but there are other recipes that you may wish to make again and again.
Oddly enough, the recipes you may wish to make most are the 'Failed Recipes'. They can be made using ANY ingredient that cannot be made or combined into a dish successfully... but why should you waste ingredients with shipping values? Use a Weed, a piece of Junk ore or rubbish to make the six 'Failed Dishes' required as ingredients in Ultimate Curry, Finest Curry and Elli Leaves. It is during the Winter Season especially that you may value the Junk Ore and rubbish that you obtain, as weeds cannot be found in Winter. I usually save 99 in my shelf for this purpose but it is good to know that supplies of Junk Ore are inexhaustible in ANY season...
Not that I wouldn't be willing to use the most valuable items, even Mythic Stone, to create Failed Dishes to use in Finest Curry designated for Skye, but why do that when a piece of Junk Ore or Weed will have the same result?