The responses seem to assume a bike can't be too stiff. Not so. For example, touring bikes often feel dead when not loaded, disproportionate with the weight. Some racing style bikes also feel dead and unresponsive after an hour or so, in spite of light weight.
Super stiff often feels alluring and responsive in the parking lot, but it can really suck out on the open road. Your legs turn to lead and you struggle to stay on top of your gear where another bike may still feel eager to go even when you are dead tired.
On metal bikes (carbon bikes are too small for me), I look for some springiness when twisting between the bars and seat tube, but if it has an exaggerated shivering that stems from my nervous unfamiliarity, it is probably too flexy. I also look for a small amount of frame springiness when pressing on one pedal with the brakes locked (about as hard as I would under hard seated riding), but not to where it feels sloppy. The movement should feel linear in relation to force. If it feels dead stiff in every direction, it won't work for me.
Unfortunately, those tests are subjective and require personal experience.