It depends on the animal causing the damage.
I don't think cats dig holes, but they leave little piles of dirt-covered poop. I use sections of rose canes in my veggie garden to keep Sam the Cat from using it as a toilet. He doesn't care for the thorns in his tootsies.
If it is skunks, opossums, or raccoons, they are most likely digging for grubs. Get rid of the grubs, and the problem eventually solves itself. They sell nematodes that kill grubs.
If you really just want to keep animals out of your garden in general, a dog might do the trick, but then you'd have a dog in your garden!

In my experience, it's the skunks. The possums usually aren't the problem, but I agree that using benefical nematodes to eliminate the grubs is the best answer.

There was a guy down the street who was allowing his new full-grown Rottweiler to poop in my rock garden. After it happened twice I went to the store and bought a big bottle of cayenne pepper powder and some black pepper as well for good measure. I sprinkled them around the garden and that stupid dog hasn't used it for an outhouse since. I don't know if that would work for whatever is bothering your plants, but it might be worth a try if nothing else does.

I still need help badly. I know the animals in my garden are skunks because I smell them very often at night. Also the holes are not covered up so they are probably not cats in those instances.

About 5 years ago I used traps and caught skunks and possums. At that time they were after my rear lawn. I suspect that the man who laid the traps were letting the animals go down the street. I was probably getting the same animals back.

Now they are primarily after my new garden of azaleas, gardenias and camellias out front with two holes in my front lawn.

Wouldn't nematodes create another problem?

Is there a deterrent contraption or whatever that you would recommend? Everyone I read about seems to have serious problems. I am getting desperate.

Here's what we do: our local Humane Society loans traps, the kind that doesn't kill the animal. We put it out at night, and in the morning there is a raccoon or a skunk in it. Then we call the HS and they come get the animal. We go to the HS and get the cage back. They can handle skunks too.

We mulch heavily, and this increases worms in the earth. That's why the animals come, to dig for the worms.

You could check at a good nursery for products that make animals stay away. The animals are most active in the summer, and even here in our moderate climate, they are not a problem in the winter. I don't know where they go or what the eat.

Nematodes will not create a new problem in your garden, but they do take a while to work, so they are a longer term solution and will do nothing to help in the short term. Some people set up motion-detection devices that set off sprinklers. Others use cheap electric fences. As far as I know, there are no magic potions you can apply to keep them out (not even coyote or lion urine). I use bird netting on my vegetable garden to keep the oppossums from digging the whole thing up, but usually the only thing that works is to make your peace with them.

Your post does not say whether or not your yard is enclosed. If you have a fenced yard, look for access points along the bottom of your fence.

I had skunks in my backyard digging & spraying. My handyman inspected the fence around my yard and found three places of entry: two spots where the skunks had pushed the fencing apart to gain access and one spot where they had found access under my house. Closing off the entry points stopped the problem.
Shadowsmom

The problem is in the front of my house where there are no fences.

I was thinking of laying out some chicken wire or Tenax on the ground in my plant beds where the problem is. Does anyone have any pro or con comments on this approach? I would have to lay down the chicken wire very close to the plants since the skunks are digging at the roots very close to the plants.