Is there a policy either allowing or disallowing listing our Twitter and/or Instagram handles in a signature?
Well, one could conceivably see that allowing social media handles in sigs could be considered tacit approval of the contents of that Facebook/Twitter account. I know that the board and its parent company could always list a disclaimer, but it can still lead to a connection in someone’s mind.
If I’m the administration (which they have repeatedly told me that I’m not), I don’t allow it. Too much of a risk for bad publicity if someone ends up being a nutjob.
You mean when. Have you been to the Pit? 
In my experience, most people have View Sigs turned off, so it won’t do a whole lot of good to have your handles there. The User Control Panel has slots for various defunct or seldom used messaging services (like AIM) that will show up in your profile, but nothing for more modern services like Twitter, FB, or Instagram.
Best bet, IMO, is to petition TPTB to change those slots to ones people currently use.
In the past, we’ve stated it’s fine for people to link to their social media things. It’s not a big deal. We reserve the ability to intercede if there is something inappropriate.
We also have a practice of only including sig one time per thread, but with the implementation of the new theme and it’s many bugs, that feature seems to auto turn on for lots of folks so we’ve relaxed much of the enforcement of that.
In addition, signature blocks shouldn’t be obnoxious.
TLDR: It’s not a big deal. Don’t be a dumbass.
Plus, of course, many people will use the same screenname on multiple sites.