Port 3306 appears to be closed on my Ubuntu server

The problem was that the server was listening internally only.

Removing the line bind-address 127.0.0.1 from /etc/mysql/my.cnf solved the issue.

Newer versions of Ubuntu (≥16.04) may have this line in /etc/mysql/mysql.conf.d/mysqld.cnf.

My suggestion, if you are sure that the ports are closed (I find it weird for a VPS to have that port closed) is to change the configuration file of MySQL to use another.

Simply open the configuration file in the terminal, sudo nano /etc/mysql/mysql.conf, and look for the [mysqld] section. In it, look for the line that reads port = 3306. Change it to another port not used and save the file.

Then simply either restart the VPS or restart the service, like sudo service mysql restart.

Just to note that, if the file mysql.conf is not in the one I mentioned above it can be in this other places:

/etc/my.conf
/etc/my.cnf
/etc/mysql/my.conf

And if the service command does not work, then do this:

sudo /etc/init.d/mysql restart

If the problem persists then in my case I would check iptables (I would actually delete everything in iptables just to start fresh if this could be an option) or any other firewall-enabled option.

Since they are VPS, I would also check the VPS Control Panel to see if it has any option that can block ports.

Apart from that, I would run nmap on the VPS to see what ports you have opened. You need to run it from outside the VPS to see what ports they have opened.

netstat -tuplen is also a good idea to see what opened ports you have on the server and which ones are in LISTEN mode.

I use this method:

sudo ufw status
sudo ufw allow XXXX/tcp
# use a port other than the default/predictable 3306
# work outside, and close the door when you are done
sudo ufw deny XXXX/tcp

No need to change other configuration files, and no extra ports opened by default.