The Bridgeburners ascended via the power of a Tanno Spiritwalker's song. If memory serves, Fiddler saved a little girl from muggers somewhere in Seven Cities. That little girl's family was headed by Kimloc, which that book implies was one of the more powerful Tanno Spiritwalkers. The song was sung into a shell, which was later broken in a battle, as a shaved knuckle(knuckles being a Malazan soldier game of chance, and shaved ones your ace in the sleeve way to cheat). That break awoke the Tanno song, calling back all the dead Bridgeburners to help win the day. This act caused the ascendency of the Bridgeburners.

Gruntle's Ascendency is complicated. When GotM starts, Fener, the Boar, is god of War. Trake, the Tiger, is a First Hero, a Soletaken, and an Ascendent, but all assume he's lost his ability to semble back into human form. Through a bunch of plot I won't spoil, Trake rises in prominence and picks Gruntle as his mortal sword. Gruntle undergoes his transformation in a horrendous siege occuring in the Pannion Domain war.

Finally, one last ascendency should be mentioned. Itkovian, the Grey Swords Shield Anvil, ascends by assuming a massive burden.

So basically, in addition to ascending via power, or conquest, we have ascension by gift and compassion, too.

I agree with the answer above, I just wanted to add further examples of Ascendency:

Ascendency via song:

Fiddler saved a little girl from muggers somewhere in Seven Cities. That little girl's family was headed by Kimloc, which that book implies was one of the more powerful Tanno Spiritwalkers. The song was sung into a shell, which was later broken in a battle, as a shaved knuckle(knuckles being a Malazan soldier game of chance, and shaved ones your ace in the sleeve way to cheat). That break awoke the Tanno song, calling back all the dead Bridgeburners to help win the day. This act caused the ascendency of the Bridgeburners.

Ascendency via deep trauma:

Gruntle's Ascendency is complicated. When GotM starts, Fener, the Boar, is god of War. Trake, the Tiger, is a First Hero, a Soletaken, and an Ascendent, but all assume he's lost his ability to semble back into human form. Through a bunch of plot I won't spoil, Trake rises in prominence and picks Gruntle as his mortal sword. Gruntle undergoes his transformation in a horrendous siege occuring in the Pannion Domain war. >!

Ascendency via compassion:

Finally, one last ascendency should be mentioned. Itkovian, the Grey Swords Shield Anvil, ascends by assuming a massive burden. >!