Welcome to Tropical Fish Keeping forum.
First question is, by "Emerald" catfish do you mean Corydoras aeneus or Brochis splendens? Both are often seen in stores under common names like Emerald Cory or Green Cory, etc. We have photos of each in our profiles, click on the shaded name to see that fish's profile; the profile text also points out the differences between them.
Brochis is significantly larger that the Cory (C. aeneus) that closely resembles it, and I would not keep Brochis in a 10g. A trio of C. aeneus can work if you are regular with maintenance (weekly partial water changes) and live plants are present. I would also add a couple neons, they will be better in a larger group. The neons and the cory are shoaling fish, meaning they live in large groups and will always be "happier" and thus healthier in groups. Six or seven neons and three corys can work.
We have extensive fish profiles here, so no need to go elsewhere.
Second tab from the left in the blue bar across the top takes you there. If the scientific or common name used in the profile is used identically in a post it will shade and you can click that for the profile.