Each major expansion of Final Fantasy XIV has featured its own set of mounts as rewards for its Extreme trials. For Heavensward, those are the Lanner mounts.
There are seven bird mounts in various colors for seven different Extreme trials, and they're all earned in more or less the same way. There is one Lanner mount that won't be covered in this article, as it isn't technically part of the set: the Wondrous Lanner, which can be purchased with Faux Leaves.
Below, we'll go over which trial you'll need to unlock and complete for each Lanner in the original set, as well as how to get the Firebird mount at the very end. If you've finished Heavensward, all of these trials should be unlockable. Each of them can be completed unsynced, which should help your farming go by faster.
All Lanner mounts in FFXIV
Rose Lanner
The Rose Lanner Whistle is obtained by defeating Ravana in Thok ast Thok (Extreme). To unlock the fight, you'll first need to complete the quest "Gods of Eld" from Torsefers. He's at The Pillars (X: 11.7, Y: 11.5), which is in Foundation. After that, complete the quest "Thok Around the Clock" from Unukalhai in The Rising Stones (X: 6.1, Y: 5.2). To fight Ravana, you'll need to be at least level 60, with an item level of 175. Each time you defeat him, you'll receive one Hive Totem.
White Lanner

To get the White Lanner Whistle, players first have to get through Bismarck in The Limitless Blue (Extreme). You'll unlock this one the same way as you would Thok ast Thok, only this time you'll take the quest titled "The Diabolical Bismarck" from Unukalhai. Bismarck's Extreme trial requires players to be at least level 60 and item level 165. Each completion rewards one Expanse Totem.
Warring Lanner
For the Warring Lanner Whistle, you'll be taking on Sephirot in Containment Bay S1T7 (Extreme). After finishing "Gods of Eld", take the quest "When the Bough Wakes" from Unukalhai to unlock the regular Containment Bay S1T7. Once you've completed it, speak with him again in The Rising Stones and take the quest titled "A Fiendish Likeness", which will unlock the Extreme version of the trial. To access it, you need to be level 60 and have an item level of at least 205. You'll receive one Fiend Totem for every win.
Sophic Lanner
The Sophic Lanner Whistle is obtained from Containment Bay P1T6 (Extreme). To unlock this trial, you'll need to complete everything required to unlock Containment Bay S1T7 first. After that, complete the quest titled "The Fate of Stars" from Unukalhai, then accept "Balance Unto All" from him to unlock the regular Containment Bay P1T6. Once you've defeated Sophia for the first time, you'll be able to accept the quest "A Deific Simulacrum" from him, which will then unlock the Extreme version. A level of 60 and an item level of 235 are required for this one. You'll earn one Goddess Totem for each successful run.
Demonic Lanner
To get the Demonic Lanner Whistle, you'll need to defeat Zurvan in Containment Bay Z1T9 (Extreme). Assuming you've unlocked Containment Bay P1T6, you should be able to accept the quest titled "The Last Pillar to Fall" from Unukalhai. This will unlock the regular version of Containment Bay Z1T9. After completing it, speak to Unukalhai again and take the quest "A Demonic Duplicate" to unlock Containment Bay Z1T9. You'll need to be level 60 and have an item level of 250. Each completion of this extreme trial will get you one Demon Totem.
Dark Lanner

The Dark Lanner Whistle drops from The Minstrel's Ballad: Nidhogg's Rage. To unlock this trial, talk to Alys in Mor Dhona (X: 21.7, Y: 8.6) and take the quest titled "Nidhogg's Rage". You'll need to have a minimum level of 60, and an item level of 220. Each time you defeat Nidhogg in this trial, you'll also receive one Horde Totem.
Round Lanner
For the Round Lanner Whistle, players will face King Thordan and His Knights Twelve in The Minstrel's Ballad: Thordan's Reign. Alys gives you the quest for this one as well: "Thordan's Reign". After unlocking the trial, you can beat the snot out of Thordan again to your heart's content. You'll need to be at level 60 with an item level of 190 to do it, though. Every success will earn you a Heavens' Ward Helm Fragment.
What to do if a mount won't drop
While it's a game of chance if you want to get each mount as a drop, there's also a failsafe method of getting each one; it will just take more work. Each time you complete one of the Extreme trials above, you'll be rewarded with a single totem, 99 of which can be exchanged for the trial's corresponding mount. You'd have to have pretty bad luck to complete any of these trials 99 times without obtaining your Lanner, but it's possible. Once you've got 99 of any of them, you can talk to Bertana in Idyllshire (X: 5.9, Y: 5.2) to exchange them for your bird of choice.
Firebird

Once you've managed to obtain all seven Lanner mounts, getting the Firebird is a cinch. Just talk to Walking Atlas in Idyllshire (X: 7.5, Y: 6.1) and complete the "Fiery Wings, Fiery Hearts" quest to get your very own Firebird Whistle.