The Birthday Massacre is a gothic synthrock band hailing from Ontario, Canada. Formed in 1999 under the name Imagica, the band is currently comprised of Chibi (vocals), Falcore (lead guitar), Rainbow (rhythm guitar, backup vocals), Rhim (drums), Owen (keyboard), and Nate Manor (bass).
The band's signature style is unique, with strong melodic synth riffs over relatively soft rock. This is coupled with dark lyrical themes often dissonating with the nostalgic-sounding melodies. The band can do hard rock as well, as shown in quite a few of their songs.
On the Mohs Scale of Rock and Metal Hardness, they range anywhere from a 2.5 to a 6.

Onto the examples! I'll be splitting this one into three categories as a result of the tremendous variation in their songs. These categories will be as follows: Instrumental Interludes, Soft Rock, and Hard Rock.

Instrumental Interludes

Red
Black
Nowhere
Trinity

Soft Rock

Pale
One
Midnight
Happy Birthday (their signature song and where they get their name from!)
Video Kid
To Die For

Hard Rock

Blue (swings around a LOT, so I could have put it in both Soft and Hard)
In the Dark
Down
Pins and Needles
Counterpane

Wow, I'm surprised you listened to me gush about them for so long. As you can tell, they're one of my favorite bands at the moment and I hope you enjoy them too. Thank you for reading!