Continuity mistake: Oliver still has a quiver full of arrows when he's making his stand at the end. A shadow demon knocks him down and all of the arrows disappear instantly. They've not been knocked out of the quiver onto the ground, they've disappeared entirely.
Other mistake: During an argument, Maggie's father mentions the border wall being built to keep people from entering the country illegally. This is obviously a reference to the real-life presidency of Donald Trump, but within the show's continuity, his comment doesn't make sense: in "Resist", the penultimate episode of season two, it's revealed that President Olivia Marsdin is literally an alien from outer space, and earlier in the season in "Welcome to Earth", she signed an executive order offering aliens such as herself full citizenship if they so desired it. If she's that tolerant of extraterrestrials, it seems unlikely that she'd be so intolerant of foreign immigrants as to resort to such an extreme action in foreign policy like building a wall on the border.
Factual error: The Marine Colonel in Supergirl played by Eddie McClintock spends all of his time wearing Marine Officer's Dress Blue Bravos. Marine Dress Blues are ceremonial uniforms. On rare occasions, the Charlies or Deltas (no coat) may be worn as a work uniforms (the primary exception to this rareness being recruiters). The full uniform is certainly not worn for day to day work in the field.
Continuity mistake: Kara and William review the article William has written about hate crimes against the transgender community. The article text is shown, and it states that Yvette's attacker, Gregory Bauer, was arrested. But Gregory is not identified and arrested until Dreamer confronts him that night.