Currently we use a custom AMI (with docker and compose installed) running bamboo on bamboo's on-demand service, but we've been bitten a couple of times as Bamboo updates their protocols without notice and causes our agents to start failing. It's not an issue if you use their official AMIs, but maintaining your own is not as easy as you'd hope.
That said, I have had some problems with the support channels at CodeShip. I know I've just been a free user evaluating the service and am probably not a high priority, but my experience hasn't been positive.
After activating Docker builds for my account, I was unable to get a test build working. It turns out that this was because of an unusual bug with service names (if I named my service `web`, it would attempt to load up what looked like an internal codeship container instead of the one specified in my config), but I didn't know that at the time.
My first support request was simply ignored. After a couple more months I decided to try a build again, but the bug was still affecting me. My second support request was dismissed as a temporary problem with the docker build services ("running out of resources") which seemed unlikely given the circumstances.
Through trial and error I discovered the service naming issue and followed up on my support ticket to report it. No response to that, either.
All in all I came away thinking it was a pretty nice service with pretty lousy support.