There's been a few attempts. I see the occasional "Zie/Zir" or "Xe/Xim", both with a "zed" sound at the beginning.
But I think you need to rethink how you use "They" every day.
Say you are talking to a co-worker, or a colleague, and they are talking about a problem with their boss. you might say, "Why not talk to your HR rep and see what they say about it." (or you might not, depending on your opinion of HR.
We all subconsciously use singular they every day of our life, all the time.