Designing circuit involving solenoid
You can, in general, detect either condition by monitoring the output current from the driver. If the current is much less than expected a short time after turning it on you know there is an open circuit failure. If the current rises above what is expected then there may may be a short and your circuit can act appropriately to protect the driver.

If you need to detect the conditions with the output off you can pass current (either a small amount that won't cause the solenoid to stick or brief pulses) through the solenoid and monitor the response. Note that you still should deal with the short circuit condition to protect the driver in case the short arises while the solenoid is energized.