Currently in the ground: parsnips, peas, radishes, broccoli, lettuce, spinach, scallions.
Carrots demand deep tilling or spading to be successful (kind of like parsnips). That, and even, consistant watering. Floods coupled with drought periods make them crack and taste weird. You might have better luck with those if you had a raised bed to plant them in. That way, poor drainage (flooding) wouldn't be an issue.