I'm the wrong person to make lengthy comments about grammar or structure, but reading through the first and half of the second post, I got the impression there were a couple of missing commas. And your pargraphs are fairly large, with chunks of dialogue that don't feel very 'vivid', for lack of a better term (sorry, my English isn't that great!) You did link them to some character action / emotion to spice things up, but with such large paragraphs I felt the essence of the characters was occasionally overpowered by the sheer amount of text packed into one paragraph.
In my humble opinion, the writing is good. Apart from the large paragraph and maybe a comma or two, nothing stood out as problematic. But the first entry could maybe offer a bit more of a hook to keep readers invested in the story. Bureaucracy is... well, bureaucracy. It's clear from the story blurb there are superpowers involved, but we don't know enough about the characters yet to feel invested in them. I'd suggest you kick the story off with something unique that has an impact and hints at exciting things to come.
Maybe the dialogue could be spiced up a bit with something beyond a polite chat about formalities. I didn't understand why Melody considered her wording so carefully, the conversation never seemed in danger of turning awkward. Missing information is maybe frustrating, but shouldn't be such a big deal. I just pictured two polite people having a polite chat about formalities. Maybe I missed something, but I didn't understand why the applicant was embarrassed, either.
The entire first paragraph of the second entry is dedicated to a description that doesn't feel immediately relevant to the story. With such short posts, you'll want every paragraph to have an impact so readers keep coming back for more.