Rosa says that DA Kurm was only 24. This however can't be true. He is first introduced as a DA in the episode Jake and Sophia in the second season almost 6 years previously which would have made him 18 at that time.
If (as Boyle suggests) Jake had made his friends eat a pig from front to back instead of a cake for the gender reveal party, they would have known the gender long before reaching the genitals. Male pigs are not really edible since their flesh has a strong taste of urine (although eating that kind of thing and wanting to make others eat it is of course perfectly in character for Boyle.)
Four of the commendation bars Captain Brenda Shawnks wears are "fake" bars that are not issued by the NYPD.
Santiago is wearing 2 citation bars that are not issued by the NYPD. Also, NYPD regulations require that the American Flag bar be worn as the highest citation bar - something Santiago would have known.
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.
After Hitchcock and Scully screw up the case against the burglars of the mayor's friend, the ADA insists that Amy reprimand them, which she does by placing them on a week's paid suspension. Amy is the sergeant in charge of the uniformed officers. She has no authority over Hitchcock and Scully.
Charles and the Peralta men made the green replacement cake. They would have seen the cake before it was frosted and therefore known it was green instead of blue.
At Terry's audition, he is asked what instrument he plays and is automatically hired when he says "flute" because he was the only flutist to audition. Terry had previously said they were holding auditions specifically for a flutist because the department band's previous flutist retired.
Jake holds a special mini-reveal for Amy at the end, as he says she never found out the sex of their baby due to the cake being green. Since the sex of the baby is revealed by the addition of food coloring to the cake mix, and there is no way to add pink coloring and come out with a green end product (green is a mixture of yellow and blue), she would have known the food coloring was blue and therefore the baby was a boy.
Jake's dad states that the plural of "stewardess" is "stewardi." It is "stewardesses."