I'd check the oil level, and check the oil for fuel. ( overfilled, or stuck float in carb, severe flooding)
Check the air filter and tubing for oil. ( could be ingesting oil)
Check the compression. ( will reveal broken ring/s, piston, or scored cylinder)
Two methods to determine oil contamination is to smell the dipstick, and also take a sheet of white notebook paper and drip a drop of oil from the dipstick on it and watch it for a minute or so. The oil will soak in a different size circle than gasoline will and can show if and how much contamination.
Also did this problem occur all of the sudden, or did it show any signs of trouble over time? Is the scooter stock or mod, and if so what mods?
White smoke is burning oil. It is either ingesting it through the intake, or it is getting past the rings due to broken ring/s, piston, scored cylinder or high level.