I've never felt weird eating alone anywhere (of course, I've never felt weird getting into conversations with people at adjoining tables, if they seem willing, either).
One advantage is, you can eat anywhere you want, i.e. if there's a restaurant or a food style that nobody with whom you normally travel likes, you can go there and have a peaceful meal

. No "there's nothing for me to eat!", or "yuck - this isn't food!", or "no, that food's going to be too spicy" (except, skip the spicy sausage appetizer at Le Cellier - trust me, it IS!).
A couple/few restaurants - California Grill, Jiko, maybe Spoodles and Narcoosees - have counters around parts of the food prep areas, where you can choose/ask to be seated if you don't want to sit at a table by yourself.
If you like talking to strangers, Biergarten and Teppan Edo seat multiple parties together at large tables. The former is a buffet, and the latter is, well, teppan-style dining - with the chef at your table, preparing your food on the grill in the center.
The only character meal I've had alone has been breakfast at Cinderella's Royal Table, but that's more because I tend to stay away from buffets than any discomfort dining solo.