Old tree stumps generally shouldn't remain in the ground in your yard; once the rest of the tree is removed, the stumps need to be dug up as well. The stumps can be removed from the property or ground up into what looks like mulch. If you have the stump ground up, the easiest option is to keep the stump grindings, but there are a couple of issues that make removing the grindings the better choice.
Use Them to Fill in the Hole
Assuming the tree was healthy and just getting old enough to become a danger (if it fell), the stump grindings are the perfect material for filling the hole left when the stump was removed. You can fill the hole with soil instead, of course, but the stump grindings provide convenient filler. The grindings will settle over the months following the stump removal, so you'll have to add soil now and again. But you won't have to worry about carting in heavy bags of soil.
Prevent Weeds and Regrowth
If you do leave the stump grindings in your yard, treat them with herbicides to ensure that any regrowth will not occur. You also want to stop weed seeds that have made their way into the pile from sprouting. Many trees can start to regrow from just a small piece of root, and if you have several pieces, you might end up with several sprouts that you then have to eradicate. Weed seeds can be dropped in or blown-in, such as on a windy day, and they can grow in the grindings/soil mixture. Treating the grindings with an herbicide prevents those problems.
Be Careful of Disease and Pests
There are times when you don't want to keep the grindings. If the tree was diseased, the stump wood might have some of those pathogens on it. Grinding up the stump and filling the hole with the infected grindings is just going to produce infected soil that could potentially infect anything you plant in it. Or, the pathogens could spread to the grass around where the stump had been. The same goes for insect pests. Those need to be removed, not buried in nutrient-rich wood bits.
An arborist can help spot disease and pests in a tree, so if you already know that the stump was infected, speak with the tree service company handling the grinding if they can remove the grindings once they're done. If you plan to keep them, talk to the company about what to look for in terms of settling and which herbicides would be best for keeping sprouts from growing.
For more information on stump grinding, contact a company near you.