My guess would be voles. We've had them for years. DH puts out mouse/rat traps baited with peanut butter. That takes care of some of them but there always seems to be more. They've cost me hundreds of dollars in plants over the years since I think their main habitat is in one of my terrace flower beds. One year we planted about 100 tulip bulbs and not a single one came up because the *&$# voles ate all of them!!! They've killed rose bushes and a wide variety of perennials.

The website below gives several suggestions on how to get rid of them. I'm sure there are more websites. I stay away from using poisons because we have dogs and lots of other wild critters around. Voles have made tracks under our lawn many times (when the snow finally melts in our backyard, the vole tracks are revealed). Once DH's traps kill a few of them, the tracks seem to go away on their own. Our yard is a hillside (fairly gentle slope) so we might not see the effects as clearly as you, but I'm sure I would notice if I had a long furrowed bump or indentation in the grass. We've also found that it's easier to keep their numbers under control if we keep the open space behind our yard cut.

Here is a link that might be useful: How to get rid of voles

"Mysterious deep holes" does not tell us much. How big around are they? Mouse sized, cat sized, or you could fall in one and be lost? How deep?

You could have anything from moles to a major sinkhole caving in, but we're going to need a little more info to help you.