I remember when Nylon bias plies first came out. After setting overnight, you not only heard the thump, thump, thump from the flat spots, you felt them.
Nowadays, most tires do not flat spot unless setting for extended periods (frequently, not even then). Generally, the better the quality of the tire, the less likely they will flat spot. Colder weather also makes tires more susceptable to flat spotting.
Flat spotting from setting long periods is more an annoyance than a problem. The two biggest problems from tires setting for long periods is rot and cracking. Tires setting on surfaces that hold moisture for long periods can cause rot where the tires have been in contact with the surface. Some surfaces, such as concrete, can also cause rot, even if seemingly dry. That is why it is recommended that one put wood between concrete and tires when setting for long periods.
Tires have to be "worked" to prevent dry rot which can cause cracking and tire failure. The flexing of a tire that is in use helps the plasticizers in the "rubber" compound work their way to the surface. When the tire is just setting, the plasticizers at the surface outgas and do not get replenished from within, making the rubber brittle and subject to cracking. That's why tires that receive little use may not last as long as tires that are frequently on the road.