I wouldn't go to a really nice restaurant alone because besides feeling uncomfortable about it, I don't care enough about food to do that. I either eat out to socialize or else to have some alone time, or else because I'm hungry. Considering those reasons for eating out, eating out alone is something I've often done. The world is full of people who work and eat lunch or dinner alone, so it wouldn't even occur to me to think about the relationship-status status issue. The world is also full of people who are in a relationship but who eat without having a companion with them. :)
I do tend to go more to either coffee shops or more casual places aimed at serving sandwiches on a plate (rather than a platter).
When it comes to finding a seat I think I'll feel more comfortable with, I look for corners, or at least places where I won't be sitting alone and staring at someone else eating, while that person stares back (even if nobody really wants to stare).
If I really wanted to see a movie badly enough I suppose I'd consider going alone; but as with eating out (more so), I go to movies for the social aspect of it. I can watch a movie alone at home.