Cold core is usually in the late or early seasons, where you get your storm to fire rather close to the center of a cold upper low. You'll hear of stacked systems. That's when the upper low center is on top of the surface low center. Often these cold core setups are close to that, but best when that surface low is not quite directly under the center of the upper low, but close. The surface low will tend to pull what moisture/juice it can back under the coldest air aloft(center of the upper low). Since it is that much colder aloft, you won't need as much heating of your surface moisture. So most cold core setups are in rather cold surface environments. It doesn't seem like things would be unstable but they are.
I myself can't stand them. Some score event after event. I seem to have chased every crappy cold core setup. My life would be fine if I never chased another one.
Another thing about cold core storms is, being as they're in close proximity to the upper low, they are "pinwheeled" around it, so the SRM on cold core storms will often be north to northwest.
I'm like H, I can't stand cold core systems. I chased one in my entire career, and that is the only CC day I can think of that didn't produce a tornado. We got an incredible supercell, while freezing to death in 40mph inflow from the north, but no tornado. Every other CC event I've ever heard of did multiple tornadoes, so blah on CC for me.