Updated for 2019 -- everything still remains the same, but added a few more details.
For any PS2 (slim or fat) with model number less than 9xxx, the simplest method is to obtain a memory card with Free MC Boot installed. The cost of such a memory card is around $10 (USD) on ebay. Once you have such a memory card, you can create new ones or update it to the most recent version (I always have one as a backup that I have stored away safely). For owners of the fat PS2 consoles, you can also...
Softmodding a (fat) PS2 in 2017
For any PS2 (slim or fat) with model number less than 9xxx, the simplest method is to obtain a memory card with Free MC Boot installed. The cost of such a memory card is around $10 (USD) on ebay. Once you have such a memory card, you can create new ones or update it to the most recent version (I always have one as a backup that I have stored away safely). For owners of the fat PS2 consoles, you can also...
Softmodding a (fat) PS2 in 2017