Sputter and shuts off

This car has not had any service (except oil and air filter) for the last 30,000 miles. Runs great, except about a year ago occasional trouble began; first every few months and now every few days. The engine will not start properly. It will pop and sputter and just not start (starter and battery are fine). Or, it will start fine, I back out of my driveway and then (thirty seconds later) the pop and sputter begins and the engine shuts off. Sometimes it will start again and sometimes it will not (and we push the car to a safe spot). If we wait some amount of time (more than a few minutes maybe a few hours or the next day) the engine starts up just fine. This problem has never occurred after the car is up and running for more than a minute. However, after driving around town (car working fine) we might park the car and thirty minutes later the engine will not start (pop and sputter). In this case we left the car overnight in a store parking lot. Sure enough, the next morning it started right up just fine.

What do you think is going on?