I usually equate black dots to caterpillar poop. I didn't think they were attracted to basil though. If you find little caterpilars on the undersides of leaves, spray with Bt. You can usually find it labeled "caterpillar killer" of some sort. It's organic, though, and won't harm beneficials.

If you only have a few, you can easily handpick the beasties.

I can't really see any black specks but I can see some very unhappy oregano. Some questions: is this plant outdoors? How long have you had it growing healthily before this happened? What is your watering policy? Is there any webbing under the leaves?

I would not spray soapy water on any plant, although some people seem to think it's a good idea. For a single plant hand picking or washing with plain water usually fixes any bug issue. But this doesn't look like a bug - more like a cultural issue.

Hi foral_uk, jean001a and loribee2, thank you for your responses!

The oregano is outdoors and only watered when the soil is dry.. it has been growing healthily for 2-4 weeks before the specks appeared about 3-5 days ago, together with a lot of black spots at first, concentrated on 1-2 leaves, although the spots disappeared and were more scattered after.. So I'm not sure if it was the one caterpillar we found (1/4 inch long), or if there are more around :(

There are also tiny holes in the leaves at scattered locations, and only a few leaves are skeleton-like :(

I haven't really spotted any more bugs, though I haven't turned the leaves over yet either...

Any chance of a good picture of the damage? I think I see one leaf that seems to show symptoms from the Four Lined Plant Bug, a pest that seems to love herbs like mint, basil, and oregano.

I'm not sure I'm seeing the same picture as everyone else. The plant I can see is very droopy and extremely sick looking - maybe from a fungus or cultural problem - but I can't see any bugs or bug faeces.