This page is dedicated to the listing of exploits which already exist used to run homebrew on the Nintendo DSi. Anyone may contribute to this list.
DSi-mode exploits
Team Twiizers had relased a DSi-Mode Exploit called Sudokuhax that loads a homebrew from the SD card in DSi Mode. The exploit requires that you have purchased the Sudoku by EA game. More details and download at: [1].
Team Twiizers also have found a DSi-Mode Exploit and have managed to use it to run DSi Mode homebrew. However it has not yet been released. More details at : [2] The additional hardware is just required to get a connection to a computer so that things like ram dumps can be created.
Wintermute has made available an open source DSi hack. The exploit works on DSi enhanced games, allowing you to run custom code from a save file. Instructions for using the exploit can be found here: [3]
If you know of DSiWare that has English-only string input,(high-scores, player name, high-scores that use username from system settings, etc) go here.
DS-mode exploits
This type of exploit is undesirable because all DSi functionality, such as usage of the cameras, is unavailable to homebrew.
Blasteh (Blasty) has posted a video on Youtube showing code being run in DS mode on the DSi using Fifa '08.