By the way, as far as I am concerned, using the quadratic equation is just another way of saying "complete the square".
Quadratic formula: In my opinion, this is the most general method, and likely the best overall. It will always work, and if you memorize the formula, there is no guessing about how to apply it. The formula allows you to find real and complex solutions.
Graphing: graphing the equation will only give valid results if the equation has real solutions. The solutions are located where the graph crosses the x axis. If the solutions are irrational or fractions with large denominators, this method will only be able to approximate the solutions. If you have a graphing calculator, this method is the quickest. If you don't have a calculator, it can be tedious and difficult to graph the equation, and you have to be very precise when you do it.
Completing the square: This is probably the most difficult method. I find it hardest to remember how to apply this method. Since the quadratic formula was derived from this method, I don't think there is a good reason to use completing the square when you have the formula (and are allowed to use it).
Factoring: this is probably the easiest method for solving an equation with integer solutions. If you can see how to split up the original equation into its factor pair, this is the quickest and allows you to solve the problem in one step.