When Dodds is implicated in the murder, his alibi is that he was at a pub with his friend Soapy Carmichael. But in S2 Ep.1, "The Man Who Wasn't There," Dodds tells Lauren that Soapy Carmichael died on last Christmas Eve.
When Dodds leaves the church, DCI McDonald's car is sometimes waiting immediately outside the door and sometimes across the road. Dodds crosses the road to get in.
When DCI McDonald buys Dodds a pint in the bar, the level in the glass is well below the top before he has even had a chance to drink any. Later he takes a drink and then the beer is the same level as earlier.
McDonald states, "Toxicology found that Felix Gibney died from a massive overdose of alprazolam, which is a prescription pain medication." Alprazolam is not a pain med; it's a benzodiazepine, an anti-anxiety drug commonly known as xanax.
Dodds says that the photograph of him on the gay contact website profile is his passport photograph. This is not possible, as he is wearing glasses in the photo, which is not permitted for a passport photo.
The murders are explicitly set in the first week of November 2022, but none of the dates and days of the week mentioned in the script match with the actual calendar e.g. 5th November is a Saturday so not a "school day tomorrow" and 7th November 2022 is a Monday not a Thursday.
When Dodds was in the nightclub talking to the singer about the knife attack that scarred her face, he said that the KILLER was hiding and expected her to walk past. But the person (man/woman?) only wounded her. Therefore not a killer.
Dodds and Milena are unable to get cell service to contact Lauren at Bishop's Gate. Yet, as soon as Maeve shoots her son, Milena is able to immediately put through a call for an ambulance.