Terry states he "pulled a suspect's phone records". Phone records require a subpoena and court order for which the operator is given considerable time to comply. No police officer can just "pull" phone records based on their hunch, absent a granted subpoena.
At silent disco the only silent component is the music. Attendees would still make considerable noise (jumping, moving, talking...), albeit within regulation limits. The literal silence in the dance hall is totally inaccurate.
Jake says he wouldn't be in debt if Holt hadn't tricked him into getting car insurance. Not only is it illegal to drive without insurance (something Jake, as a cop, would know), but Jake has been shown to be in crushing debt since before he knew Holt, even stating in The Bet (2014) that his debt began when he purchased a Mustang his first day on the force.
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.
Wuntch's commendation bars lack the black "WTC" commendation, given to all NYPD officers who served on or in the wake of the 9/11 attack on the World Trade Center. Holt, Terry, and most high-ranking officers seen throughout the series have it. Wuntch has served in the NYPD at least as long as Holt, so she should have received the WTC bar.
Madeline Wuntch mispronounces the name of the capital of the nation of Ethiopia (referred to by Captain Holt correctly as Abyssinia as it was at the time) in a manner that clearly demonstrates that she had never heard it pronounced correctly.