As a single person, I recommend that you just get out on your own. Start with the movies. Even when you go with someone, you rarely talk during the actual movie. So, head out for a matinee show and just watch the movie. After a few times, you'll realize that it's actually not bad going to the movies alone. You get to sit exactly where YOU like to sit, you don't have anyone asking "what happened?", "who's that", etc, while you're watching.
Once you get comfortable with the movies, start thinking about other activities and how doing them alone can be enjoyable.
Once I started doing this, I found that most things were actually enjoyable on my own. I still love to do things with friends and such, but I no longer depend on having a companion to get out there.
I have since traveled alone. I found the most difficult stigma to overcome was dining out alone, especially at nice restaurants. To conquer that, I started going a bit earlier in the evening to avoid the rush. And I bring a book or journal and read or write down my thoughts about things.
One interesting side effect is that I tend to be more outgoing and will chat with museum staff or waiters or other tourists, etc.
As a woman, I know that there is a wariness towards solo men, but all I can say is to just keep putting yourself out there. I find that the more I talk to people about solo life, the less I fall into the "uh oh, solo man ahead, he must be a serial killer" mindset.