In short: Aspartame is considered safe by the EFSA within the acceptable daily intake (40 mg/kg body weight) — an 85 kg person would need to drink about 7 liters of diet soda per day to exceed that. The WHO classifies it as "possibly carcinogenic," but the evidence is weak with no proven causality. Compared to sugar, it's the better bet.
Sucralose is safe when cold, but becomes problematic above about 120 °C: when heated, it breaks down into carcinogenic chlorinated compounds (dioxins). Germany's Federal Institute for Risk Assessment advises against baking or frying with sucralose. E-cigarettes with sucralose are also affected. In the EU, sucralose is therefore not approved for industrial baked goods.
Bottom line: Drinking aspartame — yes. Heating sucralose — no.