What are the symptoms that are associated with sleep disturbance?

Depends. Sleep disturbances covers a wide range of problems from sleep apnea to insomnia. Most sleep disturbances have excessive fatigue or drowsiness as the primary symptoms.
Sleepiness, usually. The typical symptoms of sleep disturbance is sleepiness. Other symptoms include fatigue, irritability, memory disturbance, difficulty concentrating, depression. The most important thing is to figure out why you have the sleep disturbance: is it sleep apnea, depression, leg kicking, insomnia? You should definitely get this checked out.