We get H12's all the time. Randomly. The only suggestion we get from Heroku is to make the requests process as fast as possible. Thus, we've spent considerable amount of time going through everything we can possibly do to make all requests respond as fast as possible. I've given up. I see this as a fundamental issue with the routing system. If you are going to use Heroku for a large production deployment, H12's (and your users getting dropped connections) will become a fact of life.
There is no auto scaling. We have no idea how many dyno's we actually need. So, we over do it in order to handle peek traffic times. This must be a great money maker for Heroku. There is no incentive for them to build auto scaling into their system because that would mean they wouldn't make as much money. Yes, auto scaling is a hard problem to solve, but there should at least be a plan to start on it and there is none that I have found.
Up until someone bitched loudly, nothing was happening to fix any of this. We have an expensive paid support contract with Heroku and before this whole routing issue blew up in public, their only recommendation was to tune the app more and buy into NewRelic for ~$8k / month. We did both and found NewRelic to not give any relevant information to help us. We did a NodeTime trial for ~$49/mo and that actually helped a lot in identifying slow spots in our app. We fixed all the slow spots in our app and still see an endless stream of H12's. Regardless, it shouldn't take a public bitch slapping for a company to listen to their customers.
You log into a dyno and see a load average of 30+. Who knows if that number is accurate or how big the underlying box really is, but regardless, I can't imagine that number being good. Am I getting H12's because I'm on an overloaded box or is it because the routing system is fundamentally broken? I don't know and nobody can tell me. This is not a good position to be in.
I have heard from several sources that Heroku isn't happy being on AWS and has been wanting to migrate off AWS for a while now. So, if your hosting provider isn't happy on their hosting provider, there must be a reason for that and in the bigger picture, you the customer, is getting screwed.
Given these things, I will never recommend that a company use Heroku. It is great if you know you are going to never have more than one dyno, but if you think you are going to go into a large production system with it, it is far better to find something else. Which brings me to another rant... how come none of these other PaaS solutions are as easy as Heroku? The git deploy is seriously the one thing they got mostly right. I'd love to see someone build a layer on top of all the PaaS solutions so that I can just deploy my code to any one of them (or event multiple).