I think this totally depend on the purpose of your experiment...deionized water means all the disslved ion are removed. Distilled water is the water that all the un-evaporated impurities are removed..mili Q water is the water that has been filtered by membrane... and the water should be the distilled water before they subjected to pass the filter...
This all depends on what you want to do with the water. in terms of purity i.e free from any contamination, then milli Q may be your best choice if available. this is followed by deionised and then distilled one.