Social dancing. Argentinian tango, to be specific.
There are two factors to make shy people more comfortable. First: "everybody in the room are doing it" - being that a class, practice or milonga. Second: codigos, a game-like ritual of asking girl to dance with a stare. If you are rejected - only her knows it.
Came here to recommend swing dancing as well. I found it a big boon in university for multiple reasons, but a big one was just that it's social environment with clear rules and expectations:
- Saying "yes" to a dance with someone is a very bounded commitment: it's 3-4 minutes of their time, and consent to physical contact with the shoulder blade and hands/arms (more for embrace dances like Balboa, but still everyone comes in knowing exactly what to expect).
- The basic moves/steps are all well understood; it's pretty easy to assess the skill level of a new partner within the first ~20 seconds and find common ground in terms of what will work to do together, how much you need to stick to the script vs being novel and playful, etc.
- There isn't much unwritten etiquette, and what little there is usually communicated clearly in beginner classes or non-judgmentally at the dances themselves. As one example: don't try to "teach" on the floor; it's okay to ask for tips or feedback if you want to, but no one wants to receive unsolicited corrections.
- In most well-run scenes no one is there to hook up and any attempt to do so would be regarded as creepy and result in the person being asked to leave. But you definitely can make same- and opposite-gender friends, and often there are opportunities to meet up beforehand or afterward, or to share travel/accommodations when going to larger events.
So yeah, my experience as an awkward engineering student in a mostly-male environment at school was that dance communities (I was connected to several of them due to traveling for internships) were very much a safe space in which to acclimatize to being around—and in limited physical contact with—women, and practice the meat of socializing in terms of how to carry a conversation, tell stories in a group, listen actively, etc, without the weird pressure of everyone suspecting everyone else's motives or like, wondering if they have some kind of agenda as far as trying to "close" or if they're going to get the wrong idea, etc. This really frees people up to be extremely friendly and have fun, and when in doubt, you're literally just there to dance and enjoy the music. :)
There are two factors to make shy people more comfortable. First: "everybody in the room are doing it" - being that a class, practice or milonga. Second: codigos, a game-like ritual of asking girl to dance with a stare. If you are rejected - only her knows it.