Jake is forced to wear an old Lilith Fair t-shirt of Amy's to work because he spent the night at her place and spilled ranch dressing over all the clothing he keeps at her apartment. He is seen wearing the t-shirt until the manhunt begins, when he is suddenly sporting a plain navy blue t-shirt without having had the chance to go home and change. If it is a department-issued shirt he acquired at the precinct, it would have made more sense for him to obtain it before heading up to the detectives' bullpen, to avoid the jeering comments of his coworkers.
Captain Holt wears his Commendation breast bar above his World Trade Center Breast Bar. When the WTC bar is worn, it is worn above all other medals, but below the American Flag bar.
The blue mailbox that Jake knocks over with Big Bertha falls without any damage to the box or the sidewalk. Real mailboxes are bolted to the concrete and would not fall without tearing the concrete and/or the box's legs.
Andy Samberg is 4" taller and about 40 pounds heavier than Melissa Fumero, yet when Jake wears one of Amy's old t-shirts it fits perfectly.
Jake's apartment is described as full of horrors, including a towel that never fully dries that was there when he moved in. The previous occupant of the apartment was Gina, whom Jake sublets from, who is not nearly as disgusting in her living conditions as he is. She would not have left behind a mystery towel with disgusting supernatural abilities.
The bus crash happens at 9:32 AM. Holt informs the squad about the crash, saying it occurred 10 minutes earlier, as Amy and Jake are just arriving for work. That would make Amy 42 minutes late for work, something she would never allow to happen, and something the others would comment upon if she did.
The 9 escaped fugitives have their mugshots displayed in the briefing room for the detectives to view. After the last "fugitive" is caught and revealed to be the van driver (who was threatened into switching places with George Gessilnick), the detectives display a photo of George (disguised as the driver, taken after the crash). Since there would be no way for the squad to obtain updated mugshots of the fugitives after the crash but before they were caught, the mugshot they had for George Gessilnick should have been of the real convict, not the van driver disguised as Gessilnick.