I recently sailed on my own. I was on a 2 week cruise in Alaska. I was invited to a solo cruiser cocktail party on the first night and there was usually an invite to a hosted table for dinner. I think the only nights that didn't happen were formal nights and the last night. As another person has posted, they request you RSVP by about 4pm. The hosted tables are quite nice. Sometimes there are many people and sometimes it's just one or two others. Also, not always the same solos (and sometimes couples) so you do get the opportunity to meet several other people. It's also really nice to have an opportunity to meet some of the crew and hear their stories.
I am an (anti-social) introvert and I was travelling just after my husband passed so I was really worried about how I would get on but it was really good. For the most part, crew and passengers are very nice and friendly and I realized that if I wanted to chat with people at breakfast or lunch, I could usually find people who wanted to share a table but I could also just sit on my own and people watch and nobody bothering me. For dinner there were the hosted tables or I ate on my own or occasionally met up with people I'd met during other meals.
I guess what I'm saying is what others have already said... you won't feel like an outcast and you can be as social (or anti-social in my case) as you want.