Is it possible to install MySQL 5.5 or 5.6 on Ubuntu 16.04?

Step by step guide* to install mysql 5.5.x on Ubuntu 16.04 Xenial-Xerus. Please see this documentation

OR

See steps below:

Installing MySQL 5.5.51 on Ubuntu 16.06

1. Uninstall any existing version of MySQL

   sudo rm /var/lib/mysql/ -R
   

2. Delete the MySQL profile

   sudo rm /etc/mysql/ -R
   

3. Automatically uninstall mysql

   sudo apt-get autoremove mysql* --purge
   sudo apt-get remove apparmor
   

4. Download version 5.5.51 from MySQL site

   wget https://dev.mysql.com/get/Downloads/MySQL-5.5/mysql-5.5.56-linux-glibc2.5-x86_64.tar.gz
   

5. Add mysql user group

   sudo groupadd mysql
   

6. Add mysql (not the current user) to mysql user group

   sudo useradd -g  mysql mysql
   

7. Extract mysql-5.5.51-linux2.6-x86_64.tar.gz to /usr/local

   cd /usr/local
   sudo tar -xvf mysql-5.5.49-linux2.6-x86_64.tar.gz
   

8. Create mysql folder in /usr/local

   sudo mv mysql-5.5.49-linux2.6-x86_64 mysql
   

9. Set mysql directory owner and user group

   cd mysql
   sudo chown -R mysql:mysql *
   

10. Install the required lib package

    sudo apt-get install libaio1
    

11. Execute mysql installation script

    sudo scripts/mysql_install_db --user=mysql
    

12. Set mysql directory owner from outside the mysql directory

    sudo chown -R root .
    

13. Set data directory owner from inside mysql directory

    sudo chown -R mysql data
    

14. Copy the mysql configuration file

    sudo cp support-files/my-medium.cnf /etc/my.cnf 
    

15. Start mysql

    sudo bin/mysqld_safe --user=mysql &
    sudo cp support-files/mysql.server /etc/init.d/mysql.server
    

16. Initialize root user password

    sudo bin/mysqladmin -u root password '111111'
    

17. Start mysql server

    sudo /etc/init.d/mysql.server start
    

18. Stop mysql server

    sudo /etc/init.d/mysql.server stop
    

19. Check status of mysql

    sudo /etc/init.d/mysql.server status
    

20. Enable myql on startup

    sudo update-rc.d -f mysql.server defaults 
    

21. Disable mysql on startup (Optional)

    sudo update-rc.d -f mysql.server remove
    

22. Add mysql path to the system

    sudo ln -s /usr/local/mysql/bin/mysql /usr/local/bin/mysql
    

23. Now directly use the command below to start mysql

    mysql -u root -p 
    

PS: One needs to reboot in order for the changes to take place.

Based on a Chinese blog

Try this, First you have to remove

sudo rm /var/lib/mysql/debian-5.7.flag

then

sudo add-apt-repository 'deb http://archive.ubuntu.com/ubuntu trusty universe'

sudo apt-get update

sudo apt install mysql-server-5.6

sudo apt install mysql-client-5.6

This answer helped me figure out how to do it by downloading the .deb packages from MySQL dev site. The solution about adding the PPA did not do it for me.

It links to a blogpost: How to Install MySQL-5-6 on Ubuntu Precise Tricky but it works.