Successfully complete the indicated task to unlock the corresponding form(s):
Future Super Saiyan: Successfully complete the "Unknown History" secret mission to unlock the Future Super Saiyan transformation.
Giant: As a Namekian, visit Guru's house at Level 35. Speak to Piccolo, then Nail, then Dende, then Piccolo again to begin a fight that will unlock the Giant form. This form drains stamina, but makes you giant and gives unique moves. It is very powerful, but the damage inflicted to you will reflect in your stamina as opposed to your health; once stamina is depleted, the form drops and you will be left in stamina break status.
Golden: As a Frieza, complete the Frieza faction by training on Frieza's Spaceship with Zarbon, then Ginyu, then Golden Frieza, and finally Meta-Cooler to unlock the Golden form. To enter Frieza's Spaceship, reach Level 20 and give Appule a Med-Mix Capsule. The Golden form provides an increase to Ki blasts and speed, and changes the nature of standard Ki blasts. Note: Level 57 is the minimum requirement for the Golden form for the Frieza race.
Kaioken: After the Saiyan Saga, successfully complete Parallel Quest 08: "Invade Earth" to potentially unlock the Kaioken transformation. The mission requires you to defeat enemies while Nappa survives (Ultimate Finish required). You will then fight revived Goku. He will use the Kaioken transformation. Defeat him to have a chance of unlocking the Kaioken transformation. If you do not get it, keep completing the mission until you get it. It may also not unlock until you have progressed further in the game, past the Future Saga. You may also have to master the skill before it unlocks. Kaioken levels are based on your Ki levels. One energy bar is normal Kaioken, 3 bars is x3 Kaioken, and 5 bars is x20 Kaioken. This transformation will drain your stamina -- at x20 it will drain very quickly. This makes both your normal and Ki attacks stronger.
Potential Unleashed: Pass all five advancement tests with a "Z" rank to unlock an additional advancement test. Complete that advancement test to get the Potential Unleashed form. The transformation costs 5 Ki bars to use.
Power Pole Pro: As a human, progress through the Hercule faction by speaking with Saiyaman (do not worry about Hercule himself). Eventually he will offer you a quest to unlock the Power Pole Pro form. This form has long reach, recovers stamina even while moving, and allows for guarding while moving.
Purification: As a Majin, progress through the Buu faction until he has three kids to get the Purification form. This form transforms you into Kid Buu, overriding your moves and stats, while also draining stamina. It is recommended to use the Potential Unleashed form instead.
Super Saiyan ("Awoken" skill): Successfully complete some story missions and reach Level 30 to unlock an event at the Capsule Corp time rift with Vegeta. As a Saiyan, complete the sparing mission with Vegeta. Complete some more story missions until you reach Level 40 and unlock the Kid Buu battle story mission. After defeating Kid Buu, while Goku is charging his spirit bomb, there will be another event with Vegeta at Capsule Corporation. Complete the mission with Vegeta, then talk to Bulma, then Trunks, then go back to Vegeta for the final battle between Goku, Vegeta, and your character. They will then teach you the "Awoken" skill, allowing you to use Super Saiyan, Super Saiyan 2, and Super Saiyan 3.
Super Vegeta: Unlock Super Saiyan, then reach Level 60. You will get a notification about a new mission at Capsule Corporation. Go there and talk to Vegeta. You will have to defeat Super Saiyan God Vegeta to unlock the Super Vegeta 1 and Super Vegeta 2 forms.