Thwack has it right. Depends on the model. I have an older OT Loon 138 that is broad and stable as a barge and just as fast. Recreational models are the way to go, both with sit-in and sit on top types. Whitewater boats need to be highly maneuverable and reactive, and are as stable as a two legged chihuahua on crack. Touring kayaks are designed for speed, not stability.

One thing to remember, especially in the spirit of the original post, that life jacket is a must, and it must be on your body securely. While a recreational kayak is harder to tip, it can be a bear to flip back once you have.

Here in PA a woman drowned last year when her kayak went over. She was experienced and had on her life jacket. What she missed was that the emergency loop to remove her sprayskirt was tucked inside the boat instead of sticking out where she could grab it, and she couldn't right the boat and couldn't get out of it.