Flash player is completely dead for all operating systems as of 2021.
Install the latest version of flash player
These instructions will install the latest version of flash player for Chromium and it will also be updated automatically. In addition to installing flash player for Chromium, it will also install/reinstall Adobe flash player for Firefox, and it will also be updated automatically.
To install the latest version of flash player search the Dash (in Ubuntu releases before 17.10) or the Show Applications dashboard (in Ubuntu 17.10 or later) for Software & Updates and open the Software & Updates window. Click the Other Software tab in the Software & Updates window, and put a check mark in the checkbox to the left of where it says: Canonical Partners.
Software & Updates in Ubuntu 17.10
Click the Close button to close the Software & Updates window.
Open the terminal and type:
sudo apt remove pepperflashplugin-nonfree # remove pepperflashplugin-nonfree if it's installed
sudo apt update
sudo apt install adobe-flashplugin
sudo apt install browser-plugin-freshplayer-pepperflash # 16.04 and later
Original answer
Don't use the instructions under this heading anymore because they are obsolete. Follow the updated "Install the latest version of flash player" instructions instead.
I found something that says Google's Chrome Browser should do the job (using a "pepper-based version"), and for Chromium there is an ppa.
There is no need to add a PPA to your software sources to install Pepper Flash Player 15 in the latest version of Chromium web browser from the Ubuntu Software Center.
In Ubuntu 14.04 and onward, Pepper Flash Player can be installed using the Ubuntu Software Center.
If a more recent version of Pepper Flash Player has been released and you want to update Pepper Flash Player to the latest version, you can do this from the terminal using the following command:
sudo update-pepperflashplugin-nonfree --install
Pepper Flash Player cannot be updated with the following command anymore on 32-bit OSs since Google ended support for Chrome on 32-bit Linux in March, 2016. If you are using Ubuntu 32-bit were using Ubuntu 32-bit before March, 2016, you can update Pepper Flash Player to the latest version from the terminal using the following command:
sudo update-pepperflashplugin-nonfree:i386 --install
Either of these two commands will download the latest version of Google Chrome (the file is larger than 40MB) and then update only the Pepper Flash Player with the more recent version of Pepper Flash Player that is bundled with Google Chrome without installing Google Chrome.
If updating Pepper Flash Player to the latest version fails with an error like this:
ERROR: failed to retrieve status information from google : W: There is no public key available for the following key IDs:
1397BC53640DB551
It happens because Google changed its signing key. To fix it run the following commands:
gpg --keyserver pgp.mit.edu --recv-keys 1397BC53640DB551
gpg --export --armor 1397BC53640DB551 | sudo sh -c 'cat >> /usr/lib/pepperflashplugin-nonfree/pubkey-google.txt'
...and then try to update Pepper Flash Player to the latest version again.
Best Answer
If you have installed chromium-browser from universe apt repository, you can make Spotify Web Player work there in Chromium, if you also install Google Chrome. Apparently that brings some shared software that Spotify needs. I tested this in 18.04.