When does the holding period begin to determine long-term capital gains?

It depends on what type of investment it is. For stocks, the holding period begins the day after your purchase order is executed; for real estate, it begins the day after acquisition, which is the earlier of 1) the date title passes to you, or 2) the date you take possession and assume the burdens and privileges of ownership. As for inherited property, it is treated as a long-term gain regardless of how long you held on to it before selling it.