For the high heat involved in deep frying fish, the best oil is one with a high smoke point, such as peanut oil, canola oil, sunflower oil and safflower oil. The smoke point is the temperature at which a fat starts breaking down and smoking due to the heat. With smoke points well above 400 degrees, these oils generally maintain proper deep-frying temperatures safely without burning.
While many sources suggest straining and reusing cooking oil, it is best to discard it. Even if it did not smoke, there is an increased chance that it has started to degrade. This also means there is possibility that it contains undesirable compounds, such as trans fats.