I will add another answer here for people seeing this in 2020 on Debian, as my resolution to this problem was not present in any of the search hits found when googling the error string "mountpoint for devices not found".

**Background:**

- Debian 8.11 running on Google Cloud Platform
- Had a working Docker install 5 weeks ago with two containers running

Suddenly realised something had made the containers crash. The only remotely probable cause I could come up with was that I had deleted a parent folder on the host of which a subfolder was mapped as a volume. Another reason could be the mounting of an additional physical device. 

The end result in any case was that trying to start any docker container resulted in the error message seen in the question ("`mountpoint for devices not found`") and no reboot (and hence upgrade of the kernel) had taken place.

The steps I took to debug the issue was

1. Inspect the logs: `journalctl -xn | less`. Didn't really contain too much additional info
1. Stop the Docker daemon (`/etc/init.d/docker stop`).
1. Add a file `/etc/docker/daemon.json` where the sole content was `{"debug": true}`
1. Try restarting the docker daemon only to see it fail 
1. Inspect the logs, which would now be filled with a lot more info

These `cgroup` related errors were the ones that led to the answer:
```
Jan 13 20:17:15 dev-diffia-no dockerd[9022]: time="2020-01-13T20:17:15.964631675Z" level=warning msg="Your kernel does not support cgroup memory limit"
Jan 13 20:17:15 dev-diffia-no dockerd[9022]: time="2020-01-13T20:17:15.964654637Z" level=warning msg="Unable to find cpu cgroup in mounts"
Jan 13 20:17:15 dev-diffia-no dockerd[9022]: time="2020-01-13T20:17:15.964667575Z" level=warning msg="Unable to find blkio cgroup in mounts"
Jan 13 20:17:15 dev-diffia-no dockerd[9022]: time="2020-01-13T20:17:15.964680057Z" level=warning msg="Unable to find cpuset cgroup in mounts"
Jan 13 20:17:15 dev-diffia-no dockerd[9022]: time="2020-01-13T20:17:15.964750643Z" level=warning msg="mountpoint for pids not found"
Jan 13 20:17:15 dev-diffia-no dockerd[9022]: time="2020-01-13T20:17:15.980250151Z" level=debug msg="Cleaning up old mountid : start."
Jan 13 20:17:15 dev-diffia-no dockerd[9022]: Error starting daemon: Devices cgroup isn't mounted
```

OK, something about `cgroups` and mounting. That led me to [a workaround for a different cgroups issue](https://github.com/docker/cli/issues/2104#issuecomment-535513985) that could be applied in this case, of which the the only commands that seemed to have an effect were

1. `/etc/init.d/docker stop`
1. `cgroupfs-mount`
1. `/etc/init.d/docker start`

Now, upon starting Docker again, the logs still contained a few lines of cgroup related errors:


```
Jan 13 20:24:42 dev-diffia-no dockerd[9775]: time="2020-01-13T20:24:42.258571633Z" level=warning msg="Your kernel does not support cgroup memory limit"
Jan 13 20:24:42 dev-diffia-no dockerd[9775]: time="2020-01-13T20:24:42.258591020Z" level=warning msg="Unable to find cpu cgroup in mounts"
Jan 13 20:24:42 dev-diffia-no dockerd[9775]: time="2020-01-13T20:24:42.258937091Z" level=warning msg="mountpoint for pids not found"
```

But half of them (`blkio`, `cpuset`) were gone, and more importantly, the next line read:

```
Jan 13 20:24:42 dev-diffia-no dockerd[9775]: time="2020-01-13T20:24:42.259420798Z" level=info msg="Loading containers: start."
```

And finally 

```
Unit docker.socket has finished starting up.
```

So, basically, remounting the cgroup stuff fixed the issue. No need to reboot.