The safety of artificial sweeteners (wikipedia) is an eteremly politisized topic, and you can find someone arguing everything from "it's 100% safe" to "it's poison" but most scientists seem to agree that they are safe for human consumption, especially in the relatively low doses we ingest them. Hey, even water will kill you if you drink enough of it.
Equal/Nutra Sweet (Aspartame): Aspartame is common in diet sodas and other low-calorie foods as well as a tabletop sweetener. Administration of large amounts of Aspartame has been shown to cause increased rates of cancer in lab rats; however, studies have not found evidence that Aspartame causes cancer in humans. There has also been suspicion that Aspartame can cause headaches and other unwanted effects in humans, but multiple studies have debunked this anecdotal evidence.
Sweet and Low (Saccharin) Saccharin was the first sugar substitute, and caused an uproar when it was found to cause bladder cancer in rats. Further research by Dr. Samuel Cohen showed that the mechanism by which Saccharin causes cancer in rats is not applicable to humans because of a specific difference in urine composition. Saccharin has since been taken off the list of suspected carcinogens by the FDA and is, for the most part, considered safe, particularly in small doses.
note: In Canada, NutraSweet does not contain Aspartame, which is banned in Canada; It contains Cyclamate, which is banned in the US.
Sweet One (Acesulfame Potassium or Ace K) Ace K is relatively new, and thus there is limited information about it's safety. It was approved by the FDA but watchdog groups argue that Ace K has not been properly tested. Ace K is most often used to mitigate the unpleasant aftertaste of Aspartame by using the two together.
Splenda (Sucralose) Introduced in 1999, Splenda recently became the market leader in artificial sweeteners. Now common in some diet sodas and other low-calorie foods as well as a tabletop sweetener. Unlike the other available sweeteners, it can be heated without causing chemical decomposition and can thus be used for baking. Because Splenda is new to the market, research on it is limited; however, animal tests have shown that Splenda can cause an enlarged thymus glad in rats when administered in very large quantities. Although this effect has not been reported in humans, it remains a potential long-term health problem.