Not in any order:
Love Droid
Beyond the Spectacle of the World
With Respect Lelouch
TOY
Poker Face
Coward Mont Blanc
Lie and Diamond
Happy Synthesizer (with Luka)
Be Myself
Blue Butterfly
Matryoshka (with Miku)
Breath
Circus
Full Volume
Holography
Liberty Heart
Naname World
Raid of Glass
Eat Me
Eye Examination
Late Excuse
Panda Hero (with Luka)
Campanella
Aitai
Bad Apple
Bird's Love Song
Cutoff Line
Dance Floor
Emiria
Eraser
For My Soul
Killer Lady
Love is Certainly Fast
Mozaik Role
Rosetta
Sorry to You

This list consists of songs under the genres of pop and rock. Gumi is a Vocaloid under the title of "Megpoid." With an interesting voice, Gumi has made herself Hatsune Miku's closest rival. I prefer Gumi over Miku, but that is only my opinion. Recently, Gumi made an appearance in a 3D concert called Vocafarre, featuring new technology called AR (Augmented Reality) that was used to make the models come to life. Also included in the concert was Lawson Akiko, Tone Rion, and Aoki Lapis. Gumi performed two songs with Akikoloid, one of which is listed above (Beyond the Spectacle of the World.) Unfortunately, Gumi has not yet been seen in concert alongside Miku yet.