I have researched the topic of moles and gophers but all of the answers seem to indicate mounds of different structures. These are tunnels that are very visible from the surface and start and stop all over the yard. I don't see any entrance holes but I have a black labrador and I have seem him digging at these tunnels. This is what prompted me go outside and take a look. He is very persistent with the digging but also known to dig because of his breed. I can not determine if these are in fact moles or gophers. We have lived here for 12 years and I have never seen either one. Do rabbits have tunnels to their nests'? There are quite a few wild rabbits in the area that have come under the fence and have dug holes to set up a nesting place.

Where do you live (State) and when do you see these? Is it in the winter, spring, summer, fall etc..

Rabbits don't do tunnels. They usually don't dig much of a hole, just reusing an old ground hog tunnel. Your tunnels sound like ground moles. They show up here from time to time. Sometimes there isn't much of an exit or entrance mound. I usually get rid of them with a trap. Try to identify a longer tunnel the others branch off from. Stomp them all down, and set the trap along what you think is the main tunnel. Watch and see where the tunnels are pushed again and keep moving the trap to that area. Sooner or later you should get it. Early in the season usually you have a pair. If you don't nail them early in the season, it will be a mess.

Note, keep the dog away from the trap. You set it with the tines flush with the ground, but dogs can manage to get in trouble given a chance.