Your pic didn't make it, sorry.

I can't tell from your description if you might be describing aphids or fungus gnats (which fly.) One other thought is that you might have a caterpillar leaving poo on your plant. If you can't decide by doing an image search for these things, try again with a pic. You can edit your original post to upload it again, so it will be at the top of the discussion.

P.S. I've had this happen to one of my images before. The cause was that I previewed the image, than spent a longer than normal time editing afterward, walking away from the keyboard to do laundry or something. I think the pic may only be stored for about 10-15 minutes before being discarded if not submitted (or re-previewed.)

Thank you for the reply. I've tried to upload the photo again. I could not get a close up, so if the photo does not help, let me continue with a description. They are very tiny and it is hard to see the structure of their body, although I expect they are not poo as I picked one up on the edge of a piece of paper and it moved its legs. They do not fly and are not very mobile. They do not seem to be fungus gnats. I was expecting aphids to be green, but maybe "black bean aphids"? Thanks for your help!

Aphids come in many colors - green, white, brown, red, orange, black. Depends on the species ;-)) And there are several species that are black.

Can't confirm it is an aphid without seeing but even if it is, it is unlikely to bother the fern. Might be more of an issue with any other indoor plants you have. Fern insect problems tend to be limited to mostly scale or mealybugs.

I would wash off the plant thoroughly at the kitchen sink. This should get rid of any aphids - if that's what they are - and the fern will appreciate the shower. While you are giving it the shower, check under all the fronds carefully and on the soil surface to make sure you dislodge and remove any creatures that may be hiding out.

Aphids do come in a rainbow of colors and can match any decor, but they rarely fly. If the pests are flying, they could be the male mealies or scale (close relatives of each other). Only the females never fly. Small black flies could be a description of males. They fly and mate. Then die. The females take up residence, then go through several stages where they can lose their legs until they resemble the scale or the mealies we are all too familiar with. Some mealies have legs, some are completely immobile. Get rid of the females, the males won't be around either.

If the pests are really tiny, it could be thrips.