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.
Character mistake: When Kara breaks her arm, she says that she doesn't need to go to a hospital because the bone will heal when her powers return. She would still need a splint or a cast to ensure that the bone sets in place properly; leaving it untreated runs the risk of a physical deformity or other unforeseen complications down the road.