Aspartame is considered the most studied food additive in existence. Decades of study and decades of consumption and yet nothing conclusive found unless ingested in obscenely large amounts.
Triggers my asthma when consumed in small amounts, but that's just me. (I only realised I was drinking a sugar-free soft drink when that happened, so I don't think it's psychosomatic.)
Other commenter pointed on actual study, and study actually says that sugar soda induced much much higher rate of disease: hazard ratio of 1.21 for sugar soda vs 1.03 for artificial sweetener soda.
Note that when deciding if that is something to worry about you need to consider the base rate for the control group.
For example if I was choosing between an activity that killed 100 out of 400 people that did it and one that killed 121 out of 400 I'd pick the first. Picking the second would raise my death probability from 25% to 30.25%. That's over a 5% increase in my death probability.
On the other hand if I was choosing between something that killed 100 out of 100 million and something that killed 121 out of 100 million, I'd probably not really consider the differences in death probabilities. For the first I've got a death probability of 0.0001% and for the second I've got a death probability of 0.000121%. My death probability is only 0.000021% higher in the second activity.
In both cases the hazard ratio is 1.21, but with the second pair of events the probability of encountering the hazard is so low a 21% increase doesn't actually make enough of a difference for me to worry about it.
Not many. They're flavored water. Randos who lay down the tired bullshit Sodas all the same are morons who don't have a clue what they're talking about. The biggest negatives are likely the acidification due to the presence of carbonation interfering vitamin absorption and enamel erosion. A minor issue is the risks from potentially unsafe food colorings still found in many American beverages.