Property taxes
The government calculates its own fair market value when it assesses and taxes the value of your property. Tax assessed value is almost always below the actual fair market value of the property, but you can protest the amount if it comes in higher than what you suspect the FMV is.
Insurance claims
Your insurance company will request the fair market value of any contents it is covering. If you make a claim, it may choose to obtain a new appraisal to update the FMV using tools such as the Kelley Blue Book value for a new or used vehicle.
Eminent domain
When the government decides it needs real estate for something like a road or government building, it sometimes has the authority to acquire that property using eminent domain. When this happens, it must pay the fair market value of the property.
Technically, this situation wouldn't meet the IRS definition of fair market value because the property owner may be an unwilling seller. However, the principles of FMV still apply. The government will hire its own experts to determine the FMV, and the owner of the property can hire experts to argue the case as well.
Gifts
The IRS requires any gifts valued at more than $15,000 to be subject to the gift tax. If the gift is valued at more than that amount, a gift tax return would need to be filed. The IRS uses FMV to calculate the value of gifts.
Stock ownership
Owners of private businesses may need to calculate the value of their business. This would be done by a business appraiser who uses techniques such as discounted cash flow to do the valuation.
If you need the FMV of stocks or bonds that you own, the current market value on the stock exchange would be used. You won't need to do any sort of advanced value investing techniques to find the FMV.