Avoid aspartame if you can. The FDA has received numerous reports of seizures and other problems. It may also be a neurotoxic. Common side effects are dizziness, visual impairment, disorientation, ear buzzing, a high level of SGOT (liver enzyme), loss of equilibrium, muscle aches, episodes of high BP, headaches, mood disturbance, and fuzzy thinking.
Other people, called 'aspartame responders', can get an addiction to aspartame. Kathleen DesMaisons, PhD (an expert in addictive nutrition), believes that the taste of any sweetener, for sugar-sensitive people, evokes a beta-endorphin response in the body which will create cravings.
Saccharin: had a reputation for cancer-causing in rats because rats got bladder tumors when they were fed high amounts (equivalent to what a human would get drinking 800 cans of diet soda a day). So it's safe, in reasonable amounts. A top choice, along with sucralose and stevia.
Cyclamate: also had a cancer-causing reputation in rats. Safe in small amounts.
Acesulfame-K: same family as Saccharin, not widely available in the states.
Sucralose (splenda): the chemical alteration prevents the digestive system from "recognizing" it, so it doesn't cause the rise in blood sugar and insulin associated with sucrose, unless of course it turns out to cause an insulin rise through a conditioned response mechanism. The problem with Splenda (theoretically), is that the chemical "alteration" involves adding chlorine molecules. The situation is evolving, but apparently this choline molecule has a lot of concerns.
Erythritol: natural sugar alcohol, no GI impact, tastes great. Most promising artificial sweetener, be on the look out for companies marketing this.
Stevia: No downside since it's an herb, but it has a weird aftertaste that some people don't like.
Credits to Jonny Bowden and his "Living Low Carb" book for this info.
I drank a movie-sized medium Diet Coke while typing this.