Hello!

Those suckers from the rootstock can be a real bear to keep up with, you're right!

For the plum tree, cutting them is probably the best option.  You can purchase "sucker stop" products but these tend to have very variable success and would not be considered a natural option if that is a priority for you.  Cut the sprouts as low as possible.

For the ash tree, the easiest solution that does not involve herbicides is to have the stump ground out or removed.  Otherwise, you will need to persistently remove the sprouts whenever they begin to grow to eventually exhaust the tree's resources, a process that could take years.  Kitchen vinegar is not a good herbicide, though horticultural strength vinegar (20% acetic acid) could be used to "burn off" new sprouts rather than cutting them with pruners.  Note, though, that horticultural vinegar is extremely dangerous to the applicator, and every precaution should be followed to protect your skin, eyes, and lungs, should you choose to use it.  Other organic herbicides, including perlargonic acid and capric acid, will also "burn off" new shoots but are much safer for you to use.  These products are available at many retailers, including hardware stores and garden centers.

If you are willing to consider other herbicides, a stump killer product applied according to label instructions would also be an option for the ash tree.  Do not apply systemic herbicides to the plum suckers, though, as you will damage the desirable tree too.

Good luck!
Cordially,

John