DOMS is caused eccentric movements that you have not properly worked up to, which cycling has none of. Running could give you DOMS in your calves if you are sedentary, or if you try minimalist shoes without working up to it, but generally cardio is not going to feature much eccentric movement and people would rarely get DOMS from any endurance work. However, that doesn't mean you can't get normal soreness if you increase time or intensity too quickly.
Improper form while cycling, such as your knees not being aligned, could cause tendonitis. On a racing style bike, the crouch position can cause lower back pain. Saddle sores are a concern for longer rides. The biggest danger, by far, is cars.