What effect would there be if the brakes were locked at touchdown? I'm imagining if you had a panicky passenger sitting in the right seat who had both feet firmly on the brakes while landing. Unless you had a crosswind and needed rudder as you touched down it's possible you wouldn't be aware of it. Or if there was a malfunction causing them to be locked. How likely is it that could cause you to have a bad day? 

What if only one side was locked up (which seems worse to me)? 

I'm mainly asking about small aircraft. Would it be any different for an airliner? 

What if you were aware of it? What action would you take?
I've seen it happen many times (ask any rental place about this). You get a big bald spot cut into the tread. Sometimes it goes all the way through the tire.

If it happened on one side, you'd likely get a ground loop.