You can ask your local ag office what grows well in shade in your location.

Usually the shade that the tree provides, and the tree’s need for water, combine to make it difficult to grow grass around the base of the trunk. I have sen people use decorative stone, or wood chips to fill in the area, but I would prefer to find some ground cover that thrives in shade. Many varieties of mint like the base of a tree. The down side of a ground cover like mint is that you need to keep it under control. I clip it and use the leaves for seasoning or tea.

Mint is kind of invasive, but if your soil is right, ferns work, and so do Lillies of the Valley. The have a wonderful fragrance in April, and are green the rest of the year.

Edit: Some Fraises du Bois do OK in the shade, too.

I like the idea of moss myself. It is my goal to have an entire yard covered with moss. I don’t know what that is going to entail, other than a really great gardener, but that’s what I want.

There is a formula for moss. You have to make the appropriate substrate for it, but once it’s established there is nothing like it.

Do the trees need thinning to let more light through? There is grass seed formulated for shade, but you really have to take extra care with shade lawns, watering more, feeding, reseeding in the spring and fall, etc.

You could plant hostas.

@PandoraBoxx
What is hostas?

Hostas All different sizes and colors, flowers on a long stalk, good in shade, establish themselves well and proliferate, can be easily divided.

Some trees poison the soil around their trunk to prevent competition. Black walnuts, butternuts, and hickories produce juglone, which is toxic to a wide variety of plants.
There are other alleopathic trees. You might have to replace the soil.

Hostas grow well in shady areas, and spread in clumps, but are not invasive. They come back year after year. Ferns also grow well in shaded areas, as does vinca. I have a small, shady yard (30 ft x 60 ft) with a privacy fence on two sides. My neighbors on either side have very old elm, walnut and chokecherry trees. I have hostas along the fence and in a center bed that I planted 12 years ago, and they are quite manageable, and bloom in late summer.

It would be helpful to know what kind of trees you have, and how old they are. You may need to invest in limb trimming if they are quite old. That can make a big difference in the amount of sun that gets to the lawn. Like @worriedguy said, certain trees produce fruit that are toxic to grass, such as black walnuts. Hard leaves, like magnolia and oak, are hard to compost, and if they’re left to sit on the grass, will kill it off.

We have pear trees which we are going to have trimmed very soon.
Believe or not since their by the curb we have to have permission from the city.
They are suppose to trim the limbs for us
Thank you @PandoraBoxx.

If they’re Bradford Pears, be aware that while they are fast growing, and have a nice shape, they have a tendency to split if they get too top-heavy.

You should be able to get decent grass underneath them, if they’re trimmed up right.

I just had an arborist come to visit me and look at 4 different fruit trees that I have—and he commented that I should get rid of the grass immediately surrounding the place where the tree meets the soil. Main reason is that it is too easy to nick the bark if you are trimming around the tree and that can cause pests and diseases to get in, so he suggested some type of mulch right around the base of the tree—I like the idea about the hostas or ferns, and leaving a little bit of space for the trunk to be bare. I think if the mice and voles can’t hide in the grass while they are gnawing on the trunk that might be a good thing too.