Pardon my late reply, @samuelwest!
Barrel distortion specifically depends on the lens. It’s usually only noticeable in budget lenses.
What is more noticeable at ultra wide angles regardless of lens is “perspective distortion”. You will definitely see perspective distortion at 10mm. But whether it’s a problem is subjective. Trees leaning inward and rocks that bulge in the corners of the frame . . . no problem for me. Buildings leaning, and faces that bulge . . . well, these could be potential problems. Usually, though, these effects are what give wide angle the neat effect that we like, so I am not concerned about perspective distortion when shooting wide. And 10mm on crop-sensor camera like yours isn’t going to look out of the ordinary. If you put the 10mm on a full frame, well, then you’re getting into fish-eye territory and that I think is problematic for “normal” photography. More about perspective distortion vs barrel distortion in the Explore Course: https://www.lenspiration.com/lesson/explore-4/
Barrel distortion would only be noticeable at the wide end of the focal length spectrum. To avoid it, zoom in, or get a more expensive lens that deals with barrel distortion better, I guess.