There are no studies that indicate any long-term health risks from drinking diet soda. Diet Soda (defined as calorie free carbonated beverages sweetened with aspartame, sucralose, acesulfame-potassium, or other non-caloric or minimally caloric sweeteners) is not harmful to health, well-being, or body composition.
Current research that attempts to link diet soda with health issues did not have equal caloric consumption. Excess caloric consumption has a direct correlation with many health issues, and in fact one of the studies distinctly states:
Consequently, the previously observed diet soda–metabolic syndrome associations are generally speculated to be the result of residual confounding by other dietary behaviors, lifestyle factors, or demographic characteristics (1,2). Biological mechanisms possibly explaining these associations are few and largely focus on artificial sweeteners in beverages/foods increasing the desire for (and consumption of) sugar-sweetened, energy-dense beverages/foods (3) or disrupting consumers' ability to accurately estimate energy intake and remaining energy needs (4). Thus, diet soda consumption may result in overconsumption, increased body weight, and consequent metabolic dysfunction.
As all current research has found diet soda sweeteners to be safe per se, a tentative conclusion is that diet soda is harmless when paired with an otherwise controlled diet; many people who have bad diets consume diet soda in an attempt to do "less damage" - this results in an incorrect conclusion that diet soda leads to an unhealthy condition.
Most likely that was a survey or epidemiological research. This is research that aims to find correlations (relationships) between two variables, and sets up future studies to find which variable causes what.
There are many studies that note that Diet Soda is frequently consumed by unhealthy people, but no studies that compare people with equivalent diets. One study (Northern Manhattan Study) was able to control a fair bit of variables and found a weak correlation between diet soda and vascular events, but these differences disappeared when 'pre-existing health conditions' were considered.
When comparing diet soda against other non-caloric beverages in an intervention study, and you control the rest of the diet, no difference in weight loss is noted. This suggests that the problem is likely the habits of people who drink diet soda, rather than the diet soda itself. The Northern Manhattan study suggested that diet soda use may be correlated with disease as it its consumption was an attempt by (already unhealthy) individuals to reduce calories and take control of their health.
Soda (in general) has been linked to poor dental health numerous times, especially children. Although sucrose (sugar) plays a major role, the general acidity of diet sodas can also negatively influence dental health. Overconsumption of soft drinks without adequate dental intervention may result in cavities and yellowing of teeth, and this applies to both sugared and diet sodas; although sugared are worse in this regard.