I have never cut myself on the blades while skating.
I've run into people (mostly kids) who cut themselves while they aren't skating, because they aren't paying attention to the edges, or to roughly ground sides of the blades. E.g., they rest their skates on their laps, or are carrying them around uncovered and let the sharp edge bump into themselves. A few cut themselves during jump landings, when the free foot slides where it isn't intended. Especially if they don't believe in socks.
I sharpen my own blades. I don't often cut myself while sharpening, but I started telling people like me who are interested in sharpening their own blades to be careful of that, when I saw a few people who did.
It is in theory be very easy to cut yourself if you feel the sharp part of the edge, or rough side rails, with your finger. If you want to feel the edge or the side rails, avoiding cuts is a matter of touching the blade very lightly. If you don't know how to be careful, just don't do it. But most cuts and abrasions happen to me if I'm not paying attention, not while I am.
I was taught that minor cuts and abrasions are not a major problem if you take care of them, unless you have trouble clotting. The usual - wash them with clean water and cover them with something sterile, to avoid infection and speed up the healing. I once had a bad infection because I didn't know I had scratched my elbow on the ice, and didn't do anything to treat it.