How many times has eminent domain been used at any or all levels of
government in any year?
If "using" the eminent domain power means actually bringing a lawsuit to obtain someone's property in exchange for court determined fair compensation, then the number of times is surprisingly small.
But, if "using" the eminent domain power includes offering to purchase property in a manner backed by the knowledge that an eminent domain suit can and will be brought is the owner doesn't agree to sell the property at the government offered price, then it is used much more often.
As one data point, there were 87 lawsuits filed to condemn property using the power of eminent domain in the 2018 fiscal year in Colorado, which has roughly 1.5% of the United States population. This doesn't include federal eminent domain proceedings, which are brought in federal court, for which there are also good statistics available. This would imply an expected 5,800 case per year, nationwide, if this figure is close to the U.S. average per capita, before considering federal cases.
When the federal government seeks to exercise its power of eminent domain, the United States government files a lawsuit in U.S. District Court, and that case is classified for statistical purposes as a "real estate" case, assuming that the government is condemning real estate (which is usually the case) and not a boat or other personal property (which is the rare exception). There were 1,001 real estate cases filed by the United States government in the U.S. District Courts in the most recent year for which statistics are available, although this is definitely an overestimate because it includes some cases that are not eminent domain proceedings. For example, this category of cases also includes federal foreclosures of mortgages owned directly by the Department of Housing and Urban Development, under Title 12, Chapters 38 and 38A, of the United States Code.
But, once an eminent domain proceeding is filed, the chances that the government will successfully obtain the property, albeit, possibly for a higher price than was offered, is very great indeed. Probably 98% or more of eminent cases filed in court will result in the acquisition of the property sought by the condemning party. The condemning party's attempt to obtain the property is probably rejected outright only one or two times a year in Colorado, on average (and some of those cases involve condemnation of property by a private party such as a purportedly landlocked land owner).
The problem with not having a more extensive data set (which wouldn't be so hard to collect from the annual reports of various state and federal judicial systems that maintain records of court filings with the detail that Colorado does), is that the number of eminent domain cases filed in a year is a very "lumpy" number. When a major infrastructure campaign is underway, like a new commuter rail line or a new highway, there are lots of eminent domain proceedings commenced. But, in other years, when there aren't a lot of major infrastructure programs in the land acquisition phase of the project, the number will be far smaller.
Still, I suspect that the total is probably less than 10,000 per year at all levels of government in the entire United States.
By comparison, there are more than 50 million parcels of owner occupied residential real estate in the United States, about 2 million farms (many of which are made up of multiple parcels of land), and many, many millions of parcels of land owned by businesses and by landlords. A reasonable estimate would be that there are about 200 million parcels of land in the United States. A "parcel" of land is also a tricky unit of analysis. A parcel could be a few square feet of land, or could be multiple square miles of land. Some states automatically merge land owned by a single owner that is contiguous into a single parcel, while others do not. And, frequently an eminent domain proceeding seeks to acquire only a small part of an entire parcel of land, such as a four foot strip of many property owner's front yards adjacent to an existing road in a road expansion project, creating new parcels of land, rather than an entire existing parcel of land.
So, something on the order of 50 parcels of real estate per million are subject to eminent domain proceedings in the United States each year. Of course, this figure doesn't include voluntary purchases of land by the government and sometimes the government sells or exchanges land as well, so the net rate of land acquisitions of governments in the United States is only loosely related to this number.
This may seem low, but this is because the process is required to begin out of court, with a good faith offer to the property owner supported by an appraisal, and only goes to court if the property owner does not accept the government's offer in the face of a looming eminent domain court case, and if the government does not accept the property owner's counteroffer. In Colorado, the government also has to pay the property owner's attorney's fees if the court determined fair market value of the property is more than 130% of the government's last, best offer prior to filing suit.
As in all other kinds of civil litigation, most cases settle, and often settle before going to court in what ends up as a voluntary sale of land to the government (albeit in the shadow of the government's right to condemn the property). If 87 cases a year actually get filed in court, this is probably the culmination of 800 to 4500 efforts to acquire land where eminent domain was an option on the table.
The monetary value of these claims
The compensation paid out
This is much more difficult to determine, because easily available court statistics don't report the outcome of the cases that are filed, and because government budget and spending documents don't distinguish between voluntary land acquisition expenditures and involuntary land acquisition expenditures. Also, many government budget and spending documents don't distinguish between land acquisition expenses and construction expenses on the acquired land.
This is also not a statistic for which crude statistical sampling is very useful. This is because a very small number of very large eminent domain acquisition for owners of large or expensive parcels of property in a small percentage of cases, make up a very large share of the total amount expended by the government in exercising its eminent domain power. The average expenditure per case is probably well in excess of the 80th percentile of expenditures in all eminent domain cases.
A third barrier is that while all eminent domain cases are corralled into either state trial courts of general jurisdiction (creating just one set of statistics per state) or the U.S. District Courts (creating just one set of statistics at the federal government level), budget and spending data is reported on a per agency basis, and at the state and local level, on a per government basis with each local government reporting its financial information separately. When all special districts with eminent domain power are included, there are on the order of 100,000 local governments in the United States. So, this information has to be derived from many more sources and those sources are not as consistent about the manner in which they report this information.
At the federal level, another possible way to aggregate information would be to obtain information from the Land Acquisition Section of the United States Justice Department which handles of eminent domain proceedings brought by the United States Government (possibly excluding a few independent agencies like the Tennessee Valley Authority), regardless of the agency acquiring the land. A report on federal land acquisition practices at a very broad level by the Congressional Research Service is found here.