I think this is unlikely to be a popped blood vessel in the eye. Although these do occur, the tend to cause a large, reddened spot of discoloration under the clear part of the eye where the blood spills out. It sounds like what you have, on the other hand, is simply prominent blood vessels in the clear part of the eye.
These prominent blood vessels could be caused by a number of different conditions leading to inflammation or chronic irritation of the eye. For example, seasonal eye allergies are a common cause of prominent, irritated blood vessels. Other causes might be exposure to physical irritants, like wind or dust. Finally, some chemical exposure can cause eye irritation; this might occur from exposure to chlorine in the eyes from a swimming pool, for example.
I would suggest going to see your primary care doctor
about this issue. They will be able to examine your eyes and help determine what is going on. If this is irritation due either to a physical or chemical exposure or allergies, your doctor will be able to prescribe medications, which might be either eyedrops or oral medications, in order to reduce the inflammation.